3876:
7363:
28254:
23040:
12508:
6353:
6845:
18920:
12033:
5898:
18455:
28451:
7358:{\displaystyle {\begin{aligned}{F_{n+1} \choose F_{n}}&=A^{n}{F_{1} \choose F_{0}}\\&=S\Lambda ^{n}S^{-1}{F_{1} \choose F_{0}}\\&=S{\begin{pmatrix}\varphi ^{n}&0\\0&(-\varphi )^{-n}\end{pmatrix}}S^{-1}{F_{1} \choose F_{0}}\\&={\begin{pmatrix}\varphi &-\varphi ^{-1}\\1&1\end{pmatrix}}{\begin{pmatrix}\varphi ^{n}&0\\0&(-\varphi )^{-n}\end{pmatrix}}{\frac {1}{\sqrt {5}}}{\begin{pmatrix}1&\varphi ^{-1}\\-1&\varphi \end{pmatrix}}{1 \choose 0},\end{aligned}}}
806:
14962:
21959:
21407:
956:
28760:
12503:{\displaystyle {\begin{aligned}1+&\varphi +\varphi ^{2}+\dots +\varphi ^{n}-\left(1+\psi +\psi ^{2}+\dots +\psi ^{n}\right)\\&={\frac {\varphi ^{n+1}-1}{\varphi -1}}-{\frac {\psi ^{n+1}-1}{\psi -1}}\\&={\frac {\varphi ^{n+1}-1}{-\psi }}-{\frac {\psi ^{n+1}-1}{-\varphi }}\\&={\frac {-\varphi ^{n+2}+\varphi +\psi ^{n+2}-\psi }{\varphi \psi }}\\&=\varphi ^{n+2}-\psi ^{n+2}-(\varphi -\psi )\\&={\sqrt {5}}(F_{n+2}-1)\\\end{aligned}}}
19311:
6348:{\displaystyle {\begin{aligned}{\vec {F}}_{n}&={\frac {1}{\sqrt {5}}}A^{n}{\vec {\mu }}-{\frac {1}{\sqrt {5}}}A^{n}{\vec {\nu }}\\&={\frac {1}{\sqrt {5}}}\varphi ^{n}{\vec {\mu }}-{\frac {1}{\sqrt {5}}}(-\varphi )^{-n}{\vec {\nu }}\\&={\cfrac {1}{\sqrt {5}}}\left({\cfrac {1+{\sqrt {5}}}{2}}\right)^{\!n}{\varphi \choose 1}\,-\,{\cfrac {1}{\sqrt {5}}}\left({\cfrac {1-{\sqrt {5}}}{2}}\right)^{\!n}{-\varphi ^{-1} \choose 1}.\end{aligned}}}
10873:
1064:
72:
28099:
14444:
22569:
199:
22802:
18915:{\displaystyle {\begin{aligned}{\bigl (}{\tfrac {2}{5}}{\bigr )}&=-1,&F_{3}&=2,&F_{2}&=1,\\{\bigl (}{\tfrac {3}{5}}{\bigr )}&=-1,&F_{4}&=3,&F_{3}&=2,\\{\bigl (}{\tfrac {5}{5}}{\bigr )}&=0,&F_{5}&=5,\\{\bigl (}{\tfrac {7}{5}}{\bigr )}&=-1,&F_{8}&=21,&F_{7}&=13,\\{\bigl (}{\tfrac {11}{5}}{\bigr )}&=+1,&F_{10}&=55,&F_{11}&=89.\end{aligned}}}
19029:
2245:
15885:
14957:{\displaystyle {\begin{aligned}(1-z-z^{2})s(z)&=\sum _{k=0}^{\infty }F_{k}z^{k}-\sum _{k=0}^{\infty }F_{k}z^{k+1}-\sum _{k=0}^{\infty }F_{k}z^{k+2}\\&=\sum _{k=0}^{\infty }F_{k}z^{k}-\sum _{k=1}^{\infty }F_{k-1}z^{k}-\sum _{k=2}^{\infty }F_{k-2}z^{k}\\&=0z^{0}+1z^{1}-0z^{1}+\sum _{k=2}^{\infty }(F_{k}-F_{k-1}-F_{k-2})z^{k}\\&=z,\end{aligned}}}
20294:
20679:
19909:
19521:
8524:
848:. In the Sanskrit poetic tradition, there was interest in enumerating all patterns of long (L) syllables of 2 units duration, juxtaposed with short (S) syllables of 1 unit duration. Counting the different patterns of successive L and S with a given total duration results in the Fibonacci numbers: the number of patterns of duration
17549:
4331:
18315:
1935:
6832:
19306:{\displaystyle 5{F_{\frac {p\pm 1}{2}}}^{2}\equiv {\begin{cases}{\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {p}{5}}{\bigr )}\pm 5\right){\pmod {p}}&{\text{if }}p\equiv 1{\pmod {4}}\\{\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {p}{5}}{\bigr )}\mp 3\right){\pmod {p}}&{\text{if }}p\equiv 3{\pmod {4}}.\end{cases}}}
6568:
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1005:
population, assuming that: a newly born breeding pair of rabbits are put in a field; each breeding pair mates at the age of one month, and at the end of their second month they always produce another pair of rabbits; and rabbits never die, but continue breeding forever. Fibonacci posed the rabbit
23000:
Thus, a male bee always has one parent, and a female bee has two. If one traces the pedigree of any male bee (1 bee), he has 1 parent (1 bee), 2 grandparents, 3 great-grandparents, 5 great-great-grandparents, and so on. This sequence of numbers of parents is the
Fibonacci sequence. The number of
23670:
SadgurushiShya writes that
Pingala was a younger brother of PÄáčini . There is an alternative opinion that he was a maternal uncle of PÄáčini . ... Agrawala , after a careful investigation, in which he considered the views of earlier scholars, has concluded that PÄáčini lived between 480 and 410
17900:
23406:"For four, variations of meters of two three being mixed, five happens. For five, variations of two earlierâthree four, being mixed, eight is obtained. In this way, for six, of four of five being mixed, thirteen happens. And like that, variations of two earlier meters being mixed, seven
20110:
20498:
19728:
19334:
15671:
23328:. In particular, it is shown how a generalized Fibonacci sequence enters the control function of finite-horizon dynamic optimisation problems with one state and one control variable. The procedure is illustrated in an example often referred to as the BrockâMirman economic growth model.
18448:
16482:
2672:
8223:
23633:
Before
Fibonacci wrote his work, the sequence Fn had already been discussed by Indian scholars, who had long been interested in rhythmic patterns ... both Gopala (before 1135 AD) and Hemachandra (c. 1150) mentioned the numbers 1,2,3,5,8,13,21 explicitly " p. 100 (3d
21751:
8206:
1855:
23320:
or infinite series-parallel circuit) follow the
Fibonacci sequence. The intermediate results of adding the alternating series and parallel resistances yields fractions composed of consecutive Fibonacci numbers. The equivalent resistance of the entire circuit equals the golden
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17365:
13744:
4072:
1589:
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20857:
23905:
It is ironic that
Leonardo, who made valuable contributions to mathematics, is remembered today mainly because a 19th-century French number theorist, Ădouard Lucas... attached the name Fibonacci to a number sequence that appears in a trivial problem in Liber
23297:
factor 1.609344 for miles to kilometers is close to the golden ratio, the decomposition of distance in miles into a sum of
Fibonacci numbers becomes nearly the kilometer sum when the Fibonacci numbers are replaced by their successors. This method amounts to a
5887:
1239:
3114:
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6702:
20475:
5343:
20087:
1409:
22928:, dividing the circle in the golden ratio. Because this ratio is irrational, no floret has a neighbor at exactly the same angle from the center, so the florets pack efficiently. Because the rational approximations to the golden ratio are of the form
19705:
15517:
6368:
16247:
25610:
En 1830, K. F. Schimper et A. Braun . Ils montraient que si l'on représente cet angle de divergence par une fraction reflétant le nombre de tours par feuille (), on tombe réguliÚrement sur un des nombres de la suite de
Fibonacci pour le numérateur
16493:
20893:
15123:
6693:
22974:, the distance from the center. Sunflowers and similar flowers most commonly have spirals of florets in clockwise and counter-clockwise directions in the amount of adjacent Fibonacci numbers, typically counted by the outermost range of radii.
21537:
5767:
13097:
4928:
2376:
17747:
2240:{\displaystyle {\begin{aligned}U_{n}&=a\varphi ^{n}+b\psi ^{n}\\&=a(\varphi ^{n-1}+\varphi ^{n-2})+b(\psi ^{n-1}+\psi ^{n-2})\\&=a\varphi ^{n-1}+b\psi ^{n-1}+a\varphi ^{n-2}+b\psi ^{n-2}\\&=U_{n-1}+U_{n-2}.\end{aligned}}}
2870:
14252:
3243:
22524:
The sequence of
Pythagorean triangles obtained from this formula has sides of lengths (3,4,5), (5,12,13), (16,30,34), (39,80,89), ... . The middle side of each of these triangles is the sum of the three sides of the preceding
16874:
15537:
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16752:
14067:
13912:
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5651:
5061:
3771:. For example, the initial values 3 and 2 generate the sequence 3, 2, 5, 7, 12, 19, 31, 50, 81, 131, 212, 343, 555, ... . The ratio of consecutive terms in this sequence shows the same convergence towards the golden ratio.
2974:
15977:
13283:
2548:
23163:. The maternal grandfather received his X chromosome from his mother, and the maternal grandmother received X chromosomes from both of her parents, so three great-grandparents contributed to the male descendant's X chromosome
7617:
23126:. The male's mother received one X chromosome from her mother (the son's maternal grandmother), and one from her father (the son's maternal grandfather), so two grandparents contributed to the male descendant's X chromosome
23043:
The number of possible ancestors on the X chromosome inheritance line at a given ancestral generation follows the
Fibonacci sequence. (After Hutchison, L. "Growing the Family Tree: The Power of DNA in Reconstructing Family
21542:
2455:
1301:
17276:
16340:
10261:
8006:
1721:
8228:
7697:
1940:
15880:{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{{F_{k}}^{2}}}={\frac {5}{24}}\!\left(\vartheta _{2}\!\left(0,{\frac {3-{\sqrt {5}}}{2}}\right)^{4}-\vartheta _{4}\!\left(0,{\frac {3-{\sqrt {5}}}{2}}\right)^{4}+1\right).}
22099:
The figure shows that 8 can be decomposed into 5 (the number of ways to climb 4 steps, followed by a single-step) plus 3 (the number of ways to climb 3 steps, followed by a double-step). The same reasoning is applied
21938:
7460:
5566:
2736:
23237:, etc. (This assumes that all ancestors of a given descendant are independent, but if any genealogy is traced far enough back in time, ancestors begin to appear on multiple lines of the genealogy, until eventually a
19008:
17043:
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3848:
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9294:
1457:
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algorithm in which an unsorted list is divided into two lists whose lengths correspond to sequential
Fibonacci numbersâby dividing the list so that the two parts have lengths in the approximate proportion
20736:
11393:
5180:
20289:{\displaystyle {\bigl (}{\tfrac {13}{5}}{\bigr )}=-1:\qquad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {13}{5}}{\bigr )}-5\right)=-5,\quad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {13}{5}}{\bigr )}+5\right)=0.}
8011:
6596:
5487:
2553:
1726:
20674:{\displaystyle {\bigl (}{\tfrac {29}{5}}{\bigr )}=+1:\qquad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {29}{5}}{\bigr )}-5\right)=0,\quad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {29}{5}}{\bigr )}+5\right)=5.}
19904:{\displaystyle {\bigl (}{\tfrac {11}{5}}{\bigr )}=+1:\qquad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {11}{5}}{\bigr )}+3\right)=4,\quad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {11}{5}}{\bigr )}-3\right)=1.}
17688:
17618:
5774:
5415:
19516:{\displaystyle {\bigl (}{\tfrac {7}{5}}{\bigr )}=-1:\qquad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {7}{5}}{\bigr )}+3\right)=-1,\quad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {7}{5}}{\bigr )}-3\right)=-4.}
7692:. The matrix formed from successive convergents of any continued fraction has a determinant of +1 or â1. The matrix representation gives the following closed-form expression for the Fibonacci numbers:
10986:
1134:
12931:
11858:
2989:
22632:, and one with the fewest nodes for a given heightâthe "thinnest" AVL tree. These trees have a number of vertices that is a Fibonacci number minus one, an important fact in the analysis of AVL trees.
15992:
11928:
20351:
8519:{\displaystyle {\begin{aligned}F_{2n-1}&={F_{n}}^{2}+{F_{n-1}}^{2}\\F_{2n{\phantom {{}-1}}}&=(F_{n-1}+F_{n+1})F_{n}\\&=(2F_{n-1}+F_{n})F_{n}\\&=(2F_{n+1}-F_{n})F_{n}.\end{aligned}}}
5202:
19966:
15399:
12697:
11626:
10706:
1322:
7958:
3447:
3332:
20732:
19578:
18460:
14449:
12038:
6850:
5903:
4052:
2310:
18153:
16133:
10531:
10068:
9124:
2794:
11500:
10452:
936:
Variations of two earlier meters ... For example, for four, variations of meters of two three being mixed, five happens. ... In this way, the process should be followed in all
11246:
11105:
20347:
10370:
9853:
2539:
1926:
19962:
19574:
17544:{\displaystyle {\begin{cases}p=5&\Rightarrow p\mid F_{p},\\p\equiv \pm 1{\pmod {5}}&\Rightarrow p\mid F_{p-1},\\p\equiv \pm 2{\pmod {5}}&\Rightarrow p\mid F_{p+1}.\end{cases}}}
15029:
6591:
4472:
3284:
63:. The sequence commonly starts from 0 and 1, although some authors start the sequence from 1 and 1 or sometimes (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the sequence begins
14392:
12004:
10929:
21429:
15299:
5659:
4326:{\displaystyle \varphi ^{n+1}=(F_{n}\varphi +F_{n-1})\varphi =F_{n}\varphi ^{2}+F_{n-1}\varphi =F_{n}(\varphi +1)+F_{n-1}\varphi =(F_{n}+F_{n-1})\varphi +F_{n}=F_{n+1}\varphi +F_{n}.}
3930:
3390:
12576:
11159:
10862:
9014:
4778:
977:
the indices from present to XII (months) as Latin ordinals and Roman numerals and the numbers (of rabbit pairs) as Hindu-Arabic numerals starting with 1, 2, 3, 5 and ending with 377.
18310:{\displaystyle \left({\frac {p}{5}}\right)={\begin{cases}0&{\text{if }}p=5\\1&{\text{if }}p\equiv \pm 1{\pmod {5}}\\-1&{\text{if }}p\equiv \pm 2{\pmod {5}}.\end{cases}}}
12935:
9964:
4820:
4560:
2301:
1706:
271:
9362:
3765:
493:
336:
24492:
1450:
15455:
10791:
2803:
947:(c. 1150) is credited with knowledge of the sequence as well, writing that "the sum of the last and the one before the last is the number ... of the next mÄtrÄ-váčtta."
14324:
14102:
8934:
4407:
4368:
16761:
8872:
11631:
9738:
3587:
16667:
13921:
13766:
8813:
428:
22364:, and a sum of Fibonacci numbers that satisfies these conditions is called a Zeckendorf representation. The Zeckendorf representation of a number can be used to derive its
21221:. All these sequences may be viewed as generalizations of the Fibonacci sequence. In particular, Binet's formula may be generalized to any sequence that is a solution of a
12540:
11544:
2899:
4809:
3959:
15892:
13125:
12028:
7494:
6827:{\displaystyle \Lambda ={\begin{pmatrix}\varphi &0\\0&-\varphi ^{-1}\!\end{pmatrix}},\quad S={\begin{pmatrix}\varphi &-\varphi ^{-1}\\1&1\end{pmatrix}}.}
22018:
8547:-th Fibonacci number from the closed-form matrix formula, but with fewer redundant steps if one avoids recomputing an already computed Fibonacci number (recursion with
24354:
14439:
10571:
4647:
4611:
3979:
3868:
23233:
23196:
23159:
23122:
23087:
23052:
inheritance line at a given ancestral generation also follows the Fibonacci sequence. A male individual has an X chromosome, which he received from his mother, and a
17124:
17091:
16919:
16655:
15017:
11279:
10297:
9686:
9653:
8750:
8712:
8679:
2385:
1248:
379:
10613:
1638:
1023:
At the end of the fourth month, the original pair has produced yet another new pair, and the pair born two months ago also produces their first pair, making 5 pairs.
21778:
14991:
10733:
9765:
9389:
8646:
8593:
6563:{\displaystyle F_{n}={\cfrac {1}{\sqrt {5}}}\left({\cfrac {1+{\sqrt {5}}}{2}}\right)^{\!n}-\,{\cfrac {1}{\sqrt {5}}}\left({\cfrac {1-{\sqrt {5}}}{2}}\right)^{\!n}.}
3714:
3687:
15450:
15332:
11422:
11040:
9881:
9620:
9594:
8619:
3119:
23545:
it was natural to consider the set of all sequences of and that have exactly m beats. ... there are exactly Fm+1 of them. For example the 21 sequences when
21970:
These numbers also give the solution to certain enumerative problems, the most common of which is that of counting the number of ways of writing a given number
16641:{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{F_{k}}}=\sum _{k=1}^{\infty }{\frac {1}{F_{2k-1}}}+\sum _{k=1}^{\infty }{\frac {1}{F_{2k}}}=3.359885666243\dots }
15421:
14277:
11006:
7370:
28642:
21108:{\displaystyle F_{1}=1,\ F_{3}=2,\ F_{5}=5,\ F_{7}=13,\ F_{9}={\color {Red}34}=2\cdot 17,\ F_{11}=89,\ F_{13}=233,\ F_{15}={\color {Red}610}=2\cdot 5\cdot 61.}
18933:
16939:
9394:
3566:. He wrote that "as 5 is to 8 so is 8 to 13, practically, and as 8 is to 13, so is 13 to 21 almost", and concluded that these ratios approach the golden ratio
22728:
head showing the arrangement in 21 (blue) and 13 (cyan) spirals. Such arrangements involving consecutive Fibonacci numbers appear in a wide variety of plants.
22386:
22839:
23344:
which uses Fibonacci intervals in some of its melodies; he viewed these as the musical counterpart to the elaborate harmony evident within nature. See also
13416:
13290:
5571:
4935:
3591:
26201:
25832:
24717:
Bugeaud, Y; Mignotte, M; Siksek, S (2006), "Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers",
24584:
4491:
1020:
At the end of the third month, the original pair produce a second pair, but the second pair only mate to gestate for a month, so there are 3 pairs in all.
17895:{\displaystyle {\begin{pmatrix}F_{m+1}&F_{m}\\F_{m}&F_{m-1}\end{pmatrix}}\equiv {\begin{pmatrix}1&1\\1&0\end{pmatrix}}^{m}{\pmod {n}}.}
12702:
17132:
16255:
10079:
5069:
21388:
Generating the next number by adding 3 numbers (tribonacci numbers), 4 numbers (tetranacci numbers), or more. The resulting sequences are known as
25830:
Patranabis, D.; Dana, S. K. (December 1985), "Single-shunt fault diagnosis through terminal attenuation measurement and using Fibonacci numbers",
25768:
17634:
17564:
5348:
15666:{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{F_{2k-1}}}={\frac {\sqrt {5}}{4}}\;\vartheta _{2}\!\left(0,{\frac {3-{\sqrt {5}}}{2}}\right)^{2},}
21852:
5505:
28285:
22824:
18443:{\displaystyle F_{p}\equiv \left({\frac {p}{5}}\right){\pmod {p}}\quad {\text{and}}\quad F_{p-\left({\frac {p}{5}}\right)}\equiv 0{\pmod {p}}.}
16477:{\displaystyle \textstyle {\frac {1}{F_{2k}}}={\sqrt {5}}\left({\frac {\psi ^{2k}}{1-\psi ^{2k}}}-{\frac {\psi ^{4k}}{1-\psi ^{4k}}}\right)\!.}
12826:
11769:
3777:
2684:
11870:
3879:
Successive tilings of the plane and a graph of approximations to the golden ratio calculated by dividing each Fibonacci number by the previous
28632:
2667:{\displaystyle {\begin{aligned}a&={\frac {U_{1}-U_{0}\psi }{\sqrt {5}}},\\b&={\frac {U_{0}\varphi -U_{1}}{\sqrt {5}}}.\end{aligned}}}
4656:
28725:
21222:
15342:
12603:
11549:
10629:
9130:
1106:
23258:, it may reflect off three surfaces: the top, middle, and bottom surfaces of the two plates. The number of different beam paths that have
7867:
3484:
25343:
21746:{\displaystyle {\frac {x}{1-x-x^{2}}}=x+x^{2}(1+x)+x^{3}(1+x)^{2}+\dots +x^{k+1}(1+x)^{k}+\dots =\sum \limits _{n=0}^{\infty }F_{n}x^{n}}
20683:
3992:
8201:{\displaystyle {\begin{aligned}{F_{m}}{F_{n}}+{F_{m-1}}{F_{n-1}}&=F_{m+n-1},\\F_{m}F_{n+1}+F_{m-1}F_{n}&=F_{m+n}.\end{aligned}}}
18012:
Fibonacci numbers. In 2006, Y. Bugeaud, M. Mignotte, and S. Siksek proved that 8 and 144 are the only such non-trivial perfect powers.
10457:
1850:{\displaystyle {\begin{aligned}\varphi ^{n}&=\varphi ^{n-1}+\varphi ^{n-2},\\\psi ^{n}&=\psi ^{n-1}+\psi ^{n-2}.\end{aligned}}}
11426:
10378:
7829:{\displaystyle {\begin{pmatrix}1&1\\1&0\end{pmatrix}}^{n}={\begin{pmatrix}F_{n+1}&F_{n}\\F_{n}&F_{n-1}\end{pmatrix}}.}
28135:
25133:
24940:
24565:
24337:
24039:
24014:
23444:
15186:
11178:
110:
Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the
20298:
15128:
11287:
9785:
2485:
1872:
28566:
26194:
24427:
19913:
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5422:
4412:
23552:
are: . In this way Indian prosodists were led to discover the Fibonacci sequence, as we have observed in Section 1.2.8 (from v.1)
14331:
10375:
A similar argument, grouping the sums by the position of the first 1 rather than the first 2 gives two more identities:
28851:
25548:
22353:. This means that every positive integer can be written as a sum of Fibonacci numbers, where any one number is used once at most.
21975:
10942:
26124:- animation of sequence, spiral, golden ratio, rabbit pair growth. Examples in art, music, architecture, nature, and astronomy
22360:
distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers. This is known as
13739:{\displaystyle F_{4n}=4F_{n}F_{n+1}\left({F_{n+1}}^{2}+2{F_{n}}^{2}\right)-3{F_{n}}^{2}\left({F_{n}}^{2}+2{F_{n+1}}^{2}\right)}
3893:
1584:{\displaystyle F_{n}={\frac {\varphi ^{n}-(-\varphi )^{-n}}{\sqrt {5}}}={\frac {\varphi ^{n}-(-\varphi )^{-n}}{2\varphi -1}}.}
28576:
26100:
26042:
26024:
25996:
25974:
25955:
25715:
25636:
25488:
25326:
25286:
25207:
24967:
24649:
24603:
24194:
24063:
23965:
23929:
23898:
23626:
23538:
23475:
25754:
Yanega, D. 1996. Sex ratio and sex allocation in sweat bees (Hymenoptera: Halictidae). J. Kans. Ent. Soc. 69 Suppl.: 98-115.
220:
95:
poetry formed from syllables of two lengths. They are named after the Italian mathematician Leonardo of Pisa, also known as
28877:
28872:
28867:
23364:
21208:
9302:
276:
25439:
3406:
3289:
28796:
27001:
26187:
972:
24845:
Broughan, Kevin A.; GonzĂĄlez, Marcos J.; Lewis, Ryan H.; Luca, Florian; MejĂa Huguet, V. Janitzio; TogbĂ©, Alain (2011),
1031:-th month, the number of pairs of rabbits is equal to the number of mature pairs (that is, the number of pairs in month
45:
in which each number is the sum of the two preceding ones. Numbers that are part of the Fibonacci sequence are known as
17:
28740:
28571:
28331:
28278:
26996:
24220:
20852:{\displaystyle 5{F_{14}}^{2}=710645\equiv 0{\pmod {29}}\;\;{\text{ and }}\;\;5{F_{15}}^{2}=1860500\equiv 5{\pmod {29}}}
18115:
9969:
9020:
2763:
22790:
advanced the idea that real instances can in part be understood as the expression of certain algebraic constraints on
28720:
27011:
26070:
25597:
24393:
23823:
23254:, when a beam of light shines at an angle through two stacked transparent plates of different materials of different
11052:
5882:{\displaystyle {\vec {F}}_{0}={1 \choose 0}={\frac {1}{\sqrt {5}}}{\vec {\mu }}-{\frac {1}{\sqrt {5}}}{\vec {\nu }},}
26991:
10302:
28730:
27704:
27284:
25628:
23866:
23839:
3248:
1234:{\displaystyle F_{n}={\frac {\varphi ^{n}-\psi ^{n}}{\varphi -\psi }}={\frac {\varphi ^{n}-\psi ^{n}}{\sqrt {5}}},}
174:
and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as
17946:
Fibonacci primes with thousands of digits have been found, but it is not known whether there are infinitely many.
3961:
as a linear function of lower powers, which in turn can be decomposed all the way down to a linear combination of
3875:
3109:{\displaystyle n_{\mathrm {largest} }(F)=\left\lfloor \log _{\varphi }{\sqrt {5}}(F+1/2)\right\rfloor ,\ F\geq 0,}
928:(c. 100 BCâc. 350 AD). However, the clearest exposition of the sequence arises in the work of
28622:
28612:
28155:
22613:
16121:{\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k+1}}{\sum _{j=1}^{k}{F_{j}}^{2}}}={\frac {{\sqrt {5}}-1}{2}}.}
10936:
10932:
10886:
7622:
24617:
21303:
use the Fibonacci recursion with other starting points to generate sequences in which all numbers are composite.
11936:
997:
where it is used to calculate the growth of rabbit populations. Fibonacci considers the growth of an idealized (
27006:
25276:
24496:
22156:
sâthey are 0000, 0001, 0010, 0100, 0101, 1000, 1001, and 1010. Such strings are the binary representations of
20470:{\displaystyle 5{F_{6}}^{2}=320\equiv -5{\pmod {13}}\;\;{\text{ and }}\;\;5{F_{7}}^{2}=845\equiv 0{\pmod {13}}}
15023:
12595:
5338:{\displaystyle {F_{k+2} \choose F_{k+1}}={\begin{pmatrix}1&1\\1&0\end{pmatrix}}{F_{k+1} \choose F_{k}}}
20082:{\displaystyle 5{F_{5}}^{2}=125\equiv 4{\pmod {11}}\;\;{\text{ and }}\;\;5{F_{6}}^{2}=320\equiv 1{\pmod {11}}}
15251:
3356:
1404:{\displaystyle \psi ={\frac {1-{\sqrt {5}}}{2}}=1-\varphi =-{1 \over \varphi }\approx -0.61803\,39887\ldots .}
28735:
28637:
28271:
28205:
28128:
27790:
26164:
24914:
22636:
16487:
11113:
10796:
8940:
4749:
31:
25400:
23787:
19700:{\displaystyle 5{F_{3}}^{2}=20\equiv -1{\pmod {7}}\;\;{\text{ and }}\;\;5{F_{4}}^{2}=45\equiv -4{\pmod {7}}}
15512:{\displaystyle s(0.001)={\frac {0.001}{0.998999}}={\frac {1000}{998999}}=0.001001002003005008013021\ldots .}
12545:
9886:
4497:
1643:
28763:
27456:
27106:
26775:
26568:
24591:
23528:
23313:
being shifted. To convert from kilometers to miles, shift the register down the Fibonacci sequence instead.
15423:
equal to any of 0.01, 0.001, 0.0001, etc. lays out the first Fibonacci numbers in the decimal expansion of
11046:
8529:
3719:
1118:
433:
128:
24456:
16242:{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{F_{2k}}}={\sqrt {5}}\left(L(\psi ^{2})-L(\psi ^{4})\right)}
1416:
28745:
27632:
27491:
27322:
27136:
27126:
26780:
26760:
26159:
24352:
André-Jeannin, Richard (1989), "Irrationalité de la somme des inverses de certaines suites récurrentes",
11009:
10750:
879:("the two are mixed") and scholars who interpret it in context as saying that the number of patterns for
27461:
25815:
22787:
17913:
14282:
28917:
28912:
28627:
27581:
27204:
27046:
26961:
26770:
26752:
26646:
26636:
26626:
26462:
26154:
26142:
26121:
25947:
23533:, vol. 4. Generating All Trees â History of Combinatorial Generation, AddisonâWesley, p. 50,
22829:
22650:
22343:
19013:
13751:
8878:
7854:
4373:
4336:
117:
27486:
25779:
15118:{\displaystyle s(z)={\frac {1}{\sqrt {5}}}\left({\frac {1}{1-\varphi z}}-{\frac {1}{1-\psi z}}\right)}
8819:
6688:{\displaystyle {\begin{aligned}A&=S\Lambda S^{-1},\\A^{n}&=S\Lambda ^{n}S^{-1},\end{aligned}}}
28882:
28617:
28607:
28597:
27709:
27254:
26875:
26661:
26656:
26651:
26641:
26618:
26014:
25539:
23375:
23358:
23316:
The measured values of voltages and currents in the infinite resistor chain circuit (also called the
23036:. This is under the unrealistic assumption that the ancestors at each level are otherwise unrelated.
22340:
22111:
9691:
3569:
27466:
25341:
Adelson-Velsky, Georgy; Landis, Evgenii (1962), "An algorithm for the organization of information",
25128:"Sequence A001175 (Pisano periods (or Pisano numbers): period of Fibonacci numbers mod n)"
22595:
of two integers: the worst case input for this algorithm is a pair of consecutive Fibonacci numbers.
19076:
18188:
17374:
9883:
based on the location of the first 2. Specifically, each set consists of those sequences that start
8766:
932:(c. 700 AD), whose own work is lost, but is available in a quotation by Gopala (c. 1135):
384:
28121:
27131:
27041:
26694:
26093:
Fibonacci's Liber Abaci: A Translation into Modern English of Leonardo Pisano's Book of Calculation
25689:
24388:, Dolciani Mathematical Expositions, vol. 9, American Mathematical Society, pp. 135â136,
23439:"Sequence A000045 (Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1)"
23324:
Brasch et al. 2012 show how a generalized Fibonacci sequence also can be connected to the field of
21532:{\displaystyle F_{n}=\sum _{k=0}^{\left\lfloor {\frac {n-1}{2}}\right\rfloor }{\binom {n-k-1}{k}}.}
15242:
11507:
8595:
can be interpreted as the number of (possibly empty) sequences of 1s and 2s whose sum is
1314:
25149:
18080:) every Fibonacci number has a prime factor that is not a factor of any smaller Fibonacci number (
12513:
5762:{\displaystyle {\vec {\mu }}={\varphi \choose 1},\quad {\vec {\nu }}={-\varphi ^{-1} \choose 1}.}
4787:
3937:
28789:
28712:
28534:
27820:
27785:
27571:
27481:
27355:
27330:
27239:
27229:
26951:
26841:
26823:
26743:
24101:
23753:
23039:
22769:
22661:
22592:
22361:
18081:
18035:
18024:
17050:
12009:
25870:
Brasch, T. von; Byström, J.; Lystad, L.P. (2012), "Optimal Control and the Fibonacci Sequence",
25707:
25701:
24935:"Sequence A235383 (Fibonacci numbers that are the product of other Fibonacci numbers)"
1071:: In a growing idealized population, the number of rabbit pairs form the Fibonacci sequence. At
28374:
28321:
28253:
28080:
27350:
27224:
26855:
26631:
26411:
26338:
26117:
Fibonacci Sequence and Golden Ratio: Mathematics in the Modern World - Mathuklasan with Sir Ram
24210:
23792:
22687:
22584:
22552:
21997:
17909:
15527:
13092:{\displaystyle F_{2n}={F_{n+1}}^{2}-{F_{n-1}}^{2}=F_{n}\left(F_{n+1}+F_{n-1}\right)=F_{n}L_{n}}
11169:
7985:
6573:
6362:
4923:{\displaystyle \varphi ^{n}={\frac {F_{n}{\sqrt {5}}\pm {\sqrt {5{F_{n}}^{2}+4(-1)^{n}}}}{2}}.}
4059:
2371:{\displaystyle \left\{{\begin{aligned}a+b&=0\\\varphi a+\psi b&=1\end{aligned}}\right.}
1110:
136:
24381:
23936:
23616:
23465:
23056:, which he received from his father. The male counts as the "origin" of his own X chromosome (
22329:
sâthey are 0001, 0111, 0101, 1000, 1010, 1110. There is an equivalent statement about subsets.
10540:
4616:
4580:
3964:
3853:
28581:
28326:
28195:
28044:
27684:
27335:
27189:
27116:
26271:
25301:
23283:
23205:
23168:
23131:
23094:
23059:
22108:
21344:
17096:
17063:
16933:
16891:
14996:
14399:
11251:
10269:
9658:
9625:
8717:
8684:
8651:
7861:
351:
18008:
Fibonacci number is 144. Attila PethĆ proved in 2001 that there is only a finite number of
10588:
2865:{\displaystyle F_{n}=\left\lfloor {\frac {\varphi ^{n}}{\sqrt {5}}}\right\rceil ,\ n\geq 0.}
28692:
28529:
28298:
27977:
27871:
27835:
27576:
27299:
27279:
27096:
26765:
26553:
25837:
25296:
25173:
24900:
24862:
24824:
24736:
24685:
24613:
24529:
24367:
23872:
23844:
23685:
22765:
22698:
22608:
22336:
21756:
21423:
21419:
21411:
21306:
Letting a number be a linear function (other than the sum) of the 2 preceding numbers. The
21180:
14969:
14247:{\displaystyle s(z)=\sum _{k=0}^{\infty }F_{k}z^{k}=0+z+z^{2}+2z^{3}+3z^{4}+5z^{5}+\dots .}
10711:
10582:
10534:
9743:
9367:
8624:
8571:
5494:
3692:
3665:
3238:{\displaystyle \log _{\varphi }(x)=\ln(x)/\ln(\varphi )=\log _{10}(x)/\log _{10}(\varphi )}
1604:
1068:
1007:
27056:
26525:
25587:
23683:
Singh, Parmanand (1985), "The So-called Fibonacci Numbers in Ancient and Medieval India",
23493:
Singh, Parmanand (1985), "The So-called Fibonacci numbers in ancient and medieval India",
15534:. For example, the sum of every odd-indexed reciprocal Fibonacci number can be written as
15426:
15308:
9770:
In a similar manner it may be shown that the sum of the first Fibonacci numbers up to the
1017:
At the end of the second month they produce a new pair, so there are 2 pairs in the field.
8:
28888:
28828:
28672:
28539:
27699:
27563:
27558:
27526:
27289:
27264:
27259:
27234:
27164:
27160:
27091:
26981:
26813:
26609:
26578:
25693:
25505:
24328:"Sequence A079586 (Decimal expansion of Sum_{k>=1} 1/F(k) where F(k) is the
23294:
22588:
21228:
Some specific examples that are close, in some sense, to the Fibonacci sequence include:
21218:
21214:
16869:{\displaystyle \sum _{k=0}^{N}{\frac {1}{F_{2^{k}}}}=3-{\frac {F_{2^{N}-1}}{F_{2^{N}}}}.}
15335:
14091:
11401:
11019:
9860:
9599:
9573:
8598:
8565:
5194:
3982:
3563:
3350:
215:
183:
112:
25841:
24740:
24433:
24155:
21783:
To see how the formula is used, we can arrange the sums by the number of terms present:
21213:
The Fibonacci sequence is one of the simplest and earliest known sequences defined by a
11760:{\displaystyle \sum _{i=1}^{n}F_{i}^{2}+{F_{n+1}}^{2}=F_{n+1}\left(F_{n}+F_{n+1}\right)}
210:
connecting the opposite corners of squares in the Fibonacci tiling (see preceding image)
28782:
28602:
28513:
28498:
28470:
28450:
28102:
27856:
27851:
27765:
27739:
27637:
27616:
27388:
27269:
27219:
27141:
27111:
27051:
26818:
26798:
26729:
26080:
25897:
25853:
25370:
Avriel, M; Wilde, DJ (1966), "Optimality of the Symmetric Fibonacci Search Technique",
25353:
25258:
25106:
24828:
24752:
24726:
24290:
24282:
24151:
23745:
23407:
23341:
23337:
23303:
23287:
22917:
22756:
pointed out the presence of the Fibonacci sequence in nature, using it to explain the (
22710:
22541:
22380:
21410:
The Fibonacci numbers are the sums of the diagonals (shown in red) of a left-justified
21300:
21184:
17299:
16747:{\displaystyle \sum _{k=0}^{\infty }{\frac {1}{F_{2^{k}}}}={\frac {7-{\sqrt {5}}}{2}},}
15406:
14262:
14062:{\displaystyle F_{kn+c}=\sum _{i=0}^{k}{k \choose i}F_{c+i}{F_{n}}^{i}{F_{n-1}}^{k-i}.}
13907:{\displaystyle F_{kn+c}=\sum _{i=0}^{k}{k \choose i}F_{c-i}{F_{n}}^{i}{F_{n+1}}^{k-i}.}
12586:
Numerous other identities can be derived using various methods. Here are some of them:
10991:
10073:
Following the same logic as before, by summing the cardinality of each set we see that
7482:
6583:
4781:
4055:
841:
84:
26986:
26127:
24093:
9391:
sequences into two non-overlapping sets where all sequences either begin with 1 or 2:
5646:{\displaystyle \psi =-\varphi ^{-1}={\tfrac {1}{2}}{\bigl (}1-{\sqrt {5}}~\!{\bigr )}}
5056:{\displaystyle \varphi ^{n}=F_{n}\varphi +F_{n-1}=(F_{n}{\sqrt {5}}+F_{n}+2F_{n-1})/2}
2969:{\displaystyle n(F)=\left\lfloor \log _{\varphi }{\sqrt {5}}F\right\rceil ,\ F\geq 1.}
28702:
28503:
28475:
28429:
28419:
28399:
28384:
28229:
28210:
28170:
28098:
27996:
27941:
27795:
27770:
27744:
27199:
27194:
27121:
27101:
27086:
26808:
26790:
26709:
26699:
26684:
26447:
26096:
26095:, Sources and Studies in the History of Mathematics and Physical Sciences, Springer,
26066:
26038:
26020:
25992:
25970:
25951:
25711:
25664:
25632:
25593:
25484:
25322:
25282:
25203:
24963:
24888:
24832:
24812:
24645:
24599:
24389:
24294:
24245:
24241:
24216:
24190:
24059:
24009:"Sequence A002390 (Decimal expansion of natural logarithm of golden ratio)"
23961:
23925:
23894:
23819:
23703:
23622:
23534:
23507:
23471:
23307:
22657:
22350:
22332:
21157:
18016:
17981:
16651:
15889:
If we add 1 to each Fibonacci number in the first sum, there is also the closed form
13747:
11108:
11013:
8757:
7475:
4812:
1122:
27521:
25857:
24756:
15972:{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{1+F_{2k-1}}}={\frac {\sqrt {5}}{2}},}
13278:{\displaystyle F_{3n}=2{F_{n}}^{3}+3F_{n}F_{n+1}F_{n-1}=5{F_{n}}^{3}+3(-1)^{n}F_{n}}
28687:
28508:
28434:
28424:
28404:
28306:
28185:
28032:
27825:
27411:
27383:
27373:
27365:
27249:
27214:
27209:
27176:
26870:
26833:
26724:
26719:
26714:
26704:
26676:
26563:
26510:
26467:
26406:
26006:
25984:
25901:
25887:
25879:
25845:
25660:
25454:
25321:, vol. 1: Fundamental Algorithms (3rd ed.), AddisonâWesley, p. 343,
25098:
24804:
24744:
24673:
24515:
24505:
24274:
23953:
23698:
23502:
23255:
22725:
22693:
22537:
22533:
22365:
22157:
21351:
17985:
17707:
17057:
7612:{\displaystyle \varphi =1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}}}.}
1126:
845:
800:
42:
26515:
25769:"Growing the Family Tree: The Power of DNA in Reconstructing Family Relationships"
25089:
Freyd, Peter; Brown, Kevin S. (1993), "Problems and Solutions: Solutions: E3410",
23200:. Five great-great-grandparents contributed to the male descendant's X chromosome
22716:
22573:
21179:, which includes as a subproblem a special instance of the problem of finding the
18102:) are the only Fibonacci numbers that are the product of other Fibonacci numbers.
1057:
28465:
28394:
28180:
28008:
27897:
27830:
27756:
27679:
27653:
27471:
27184:
26976:
26946:
26936:
26931:
26597:
26505:
26452:
26296:
26236:
26056:
25292:
25223:
25169:
24955:
24896:
24858:
24820:
24698:
PethĆ, Attila (2001), "Diophantine properties of linear recursive sequences II",
24681:
24609:
24525:
24408:
24363:
23919:
23345:
23317:
23280:, there are three reflection paths, not two, one for each of the three surfaces.)
23048:
It has similarly been noticed that the number of possible ancestors on the human
22773:
22682:
Some Agile teams use a modified series called the "Modified Fibonacci Series" in
21196:
18110:
17988:, there are therefore also arbitrarily long runs of composite Fibonacci numbers.
17925:
17558:
3559:
1056:
The name "Fibonacci sequence" was first used by the 19th-century number theorist
25776:
Proceedings of the First Symposium on Bioinformatics and Biotechnology (BIOT-04)
24846:
24748:
23957:
22784:) of plants were frequently expressed as fractions involving Fibonacci numbers.
2450:{\displaystyle a={\frac {1}{\varphi -\psi }}={\frac {1}{\sqrt {5}}},\quad b=-a,}
1296:{\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}\approx 1.61803\,39887\ldots }
818:) ways of arranging long and short syllables in a cadence of length six. Eight (
79:
whose side lengths are successive Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13 and 21
28697:
28682:
28677:
28356:
28341:
28190:
28013:
27881:
27866:
27730:
27694:
27669:
27545:
27516:
27501:
27378:
27274:
27244:
26971:
26926:
26803:
26401:
26396:
26391:
26363:
26348:
26261:
26246:
26224:
26211:
25697:
25462:
24677:
24580:
23886:
23694:
23369:
23238:
22732:
Fibonacci sequences appear in biological settings, such as branching in trees,
22683:
22643:
22529:
22376:
18031:
17628:
17271:{\displaystyle \gcd(F_{n},F_{n+1})=\gcd(F_{n},F_{n+2})=\gcd(F_{n+1},F_{n+2})=1}
16936:. In fact, the Fibonacci sequence satisfies the stronger divisibility property
16335:{\displaystyle \textstyle L(q):=\sum _{k=1}^{\infty }{\frac {q^{k}}{1-q^{k}}},}
16250:
15531:
14257:
10256:{\displaystyle F_{n+2}=F_{n}+F_{n-1}+...+|\{(1,1,...,1,2)\}|+|\{(1,1,...,1)\}|}
7989:
4574:
2979:
187:
132:
124:
121:
99:, who introduced the sequence to Western European mathematics in his 1202 book
26115:
26016:
Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity
25883:
25730:
25416:
24770:
24520:
23089:), and at his parents' generation, his X chromosome came from a single parent
22356:
Moreover, every positive integer can be written in a unique way as the sum of
22037:
ways one can climb a staircase of 5 steps, taking one or two steps at a time:
28906:
28662:
28336:
28200:
28144:
27936:
27920:
27861:
27815:
27511:
27496:
27406:
26689:
26558:
26520:
26477:
26358:
26343:
26333:
26291:
26281:
26256:
26179:
26010:
25849:
25520:
24892:
24816:
24306:
23410:
twenty-one. In this way, the process should be followed in all mÄtrÄ-váčttas"
23381:
22556:
22255:
sâthey are 0000, 0011, 0110, 1100, 1111. Equivalently, the number of subsets
21991:
21355:
21233:
21172:
21147:
18009:
18005:
17691:
15302:
13755:
10747:-th Fibonacci numbers. To see this, begin with a Fibonacci rectangle of size
7976:
4650:
955:
924:
805:
203:
26138:
24596:
Connections in Discrete Mathematics: A Celebration of the Work of Ron Graham
21958:
21406:
135:
used for interconnecting parallel and distributed systems. They also appear
116:. Applications of Fibonacci numbers include computer algorithms such as the
28667:
28409:
28351:
28219:
28165:
28160:
27972:
27961:
27876:
27714:
27689:
27606:
27506:
27476:
27451:
27435:
27340:
27307:
27030:
26941:
26880:
26457:
26353:
26286:
26266:
26241:
26133:
26062:
25625:
Lindenmayer Systems, Fractals, and Plants (Lecture Notes in Biomathematics)
25458:
25314:
24545:, Richard Crandall, Carl Pomerance, Springer, second edition, 2005, p. 142.
24510:
23612:
23524:
23053:
23049:
22982:
22925:
22757:
21247:
21188:
21176:
17935:
17289:
15986:
14095:
13115:
1306:
1091:
919:
207:
179:
159:
23378: â Randomized mathematical sequence based upon the Fibonacci sequence
23334:
included the Fibonacci sequence in some of his artworks beginning in 1970.
21933:{\displaystyle \textstyle {\binom {5}{0}}+{\binom {4}{1}}+{\binom {3}{2}}}
21339:. If the coefficient of the preceding value is assigned a variable value
7992:, and one may easily deduce the second one from the first one by changing
7455:{\displaystyle F_{n}={\cfrac {\varphi ^{n}-(-\varphi )^{-n}}{\sqrt {5}}}.}
5561:{\displaystyle \varphi ={\tfrac {1}{2}}{\bigl (}1+{\sqrt {5}}~\!{\bigr )}}
2892:. This formula is easily inverted to find an index of a Fibonacci number
2731:{\textstyle \left|{\frac {\psi ^{n}}{\sqrt {5}}}\right|<{\frac {1}{2}}}
28820:
28414:
28361:
28224:
27931:
27806:
27611:
27075:
26966:
26921:
26916:
26666:
26573:
26472:
26301:
26276:
26251:
26052:
25278:
Introduction to Circle Packing: The Theory of Discrete Analytic Functions
25123:
24930:
24555:
24323:
24029:
24004:
23434:
22990:
22781:
22733:
22625:
22621:
21307:
21240:
19003:{\displaystyle F_{p-\left({\frac {p}{5}}\right)}\equiv 0{\pmod {p^{2}}}.}
17038:{\displaystyle \gcd(F_{a},F_{b},F_{c},\ldots )=F_{\gcd(a,b,c,\ldots )}\,}
14078:
in this formula, one gets again the formulas of the end of above section
10708:
or in words, the sum of the squares of the first Fibonacci numbers up to
9570:
Excluding the first element, the remaining terms in each sequence sum to
9563:{\displaystyle F_{n}=|\{(1,...),(1,...),...\}|+|\{(2,...),(2,...),...\}|}
8753:
8548:
7988:
can be derived (they are obtained from two different coefficients of the
7471:
5654:
3986:
3843:{\displaystyle \lim _{n\to \infty }{\frac {F_{n+m}}{F_{n}}}=\varphi ^{m}}
1014:
At the end of the first month, they mate, but there is still only 1 pair.
984:
966:
944:
140:
102:
25892:
25262:
22517:{\displaystyle (F_{n}F_{n+3})^{2}+(2F_{n+1}F_{n+2})^{2}={F_{2n+3}}^{2}.}
1593:
To see the relation between the sequence and these constants, note that
28263:
28068:
28049:
27345:
26956:
26058:
The Golden Ratio: The Story of Phi, the World's Most Astonishing Number
25110:
24975:
24808:
24286:
23952:, Einblick in die Wissenschaft, Vieweg+Teubner Verlag, pp. 87â98,
23384: â Infinite matrix of integers derived from the Fibonacci sequence
23331:
22899:{\displaystyle \theta ={\frac {2\pi }{\varphi ^{2}}}n,\ r=c{\sqrt {n}}}
22791:
22777:
22672:
22599:
22548:
22372:
18020:
5490:
2982:
gives the largest index of a Fibonacci number that is not greater than
25651:
Vogel, Helmut (1979), "A better way to construct the sunflower head",
24876:
24034:"Sequence A097348 (Decimal expansion of arccsch(2)/log(10))"
22107:
The Fibonacci numbers can be found in different ways among the set of
13533:{\displaystyle F_{3n+2}={F_{n+1}}^{3}+3{F_{n+1}}^{2}F_{n}+{F_{n}}^{3}}
13407:{\displaystyle F_{3n+1}={F_{n+1}}^{3}+3F_{n+1}{F_{n}}^{2}-{F_{n}}^{3}}
10581:-th Fibonacci number, and the sum of the first Fibonacci numbers with
4739:{\displaystyle F_{n}=(\varphi ^{n}-(-1)^{n}\varphi ^{-n})/{\sqrt {5}}}
3850:, because the ratios between consecutive Fibonacci numbers approaches
3653:{\displaystyle \lim _{n\to \infty }{\frac {F_{n+1}}{F_{n}}}=\varphi .}
875: 450 BCâ200 BC). Singh cites Pingala's cryptic formula
28805:
28346:
27674:
27601:
27593:
27398:
27312:
26430:
26134:
Scientists find clues to the formation of Fibonacci spirals in nature
25944:
Strange Curves, Counting Rabbits, and Other Mathematical Explorations
24731:
24250:
23948:
Beutelspacher, Albrecht; Petri, Bernhard (1996), "Fibonacci-Zahlen",
23325:
22820:
22745:
22741:
22737:
22721:
22208:
22101:
20887:(as the products of odd prime divisors) are congruent to 1 modulo 4.
17554:
9289:{\displaystyle F_{5}=5=|\{(1,1,1,1),(1,1,2),(1,2,1),(2,1,1),(2,2)\}|}
8528:
These last two identities provide a way to compute Fibonacci numbers
994:
960:
929:
152:
148:
144:
96:
25102:
24987:
24278:
22371:
Starting with 5, every second Fibonacci number is the length of the
17056:
In particular, any three consecutive Fibonacci numbers are pairwise
10872:
3545:{\displaystyle n\log _{b}\varphi ={\frac {n\log \varphi }{\log b}}.}
1063:
71:
28294:
27775:
24185:
Vorobiev, NikolaÄ Nikolaevich; Martin, Mircea (2002), "Chapter 1",
22795:
22761:
22749:
22629:
22568:
22310:. For example, out of the 16 binary strings of length 4, there are
22241:. For example, out of the 16 binary strings of length 4, there are
22142:. For example, out of the 16 binary strings of length 4, there are
2797:
1102:
198:
92:
25018:
22676:
17295:
divides a Fibonacci number that can be determined by the value of
15019:
cancel out because of the defining Fibonacci recurrence relation.
28113:
27780:
27439:
27433:
26171:
25053:
24999:
22801:
21223:
homogeneous linear difference equation with constant coefficients
6361:
th element in the Fibonacci series may be read off directly as a
2757:
998:
868:
88:
22768:
most often have petals in counts of Fibonacci numbers. In 1830,
22686:, as an estimation tool. Planning Poker is a formal part of the
22335:
was able to show that the Fibonacci numbers can be defined by a
12810:{\displaystyle {F_{n}}^{2}-F_{n+r}F_{n-r}=(-1)^{n-r}{F_{r}}^{2}}
8564:
Most identities involving Fibonacci numbers can be proved using
24127:
23251:
22816:
22753:
22675:
the original audio wave similar to logarithmic methods such as
22379:
with integer sides, or in other words, the largest number in a
22115:
20876:
are congruent to 1 modulo 4, implying that all odd divisors of
15234:{\displaystyle \psi ={\tfrac {1}{2}}\left(1-{\sqrt {5}}\right)}
15176:{\textstyle \varphi ={\tfrac {1}{2}}\left(1+{\sqrt {5}}\right)}
8714:, meaning the empty sequence "adds up" to 0. In the following,
8543:
arithmetic operations. This matches the time for computing the
1002:
975:
showing (in box on right) 13 entries of the Fibonacci sequence:
76:
26495:
24877:"On Perfect numbers which are ratios of two Fibonacci numbers"
22104:
until a single step, of which there is only one way to climb.
15985:
sum of squared Fibonacci numbers giving the reciprocal of the
11388:{\displaystyle \sum _{i=1}^{n}F_{i}+F_{n+1}=F_{n+1}+F_{n+2}-1}
10960:
5175:{\displaystyle 5{F_{n}}^{2}+4(-1)^{n}=(F_{n}+2F_{n-1})^{2}\,.}
2288:
must be the Fibonacci sequence. This is the same as requiring
867:
Knowledge of the Fibonacci sequence was expressed as early as
25249:
Pagni, David (September 2001), "Fibonacci Meets Pythagoras",
24795:
Luca, Florian (2000), "Perfect Fibonacci and Lucas numbers",
24128:"The Golden Ratio, Fibonacci Numbers and Continued Fractions"
23721:, Jodhpur: Rajasthan Oriental Research Institute, p. 101
23299:
22989:
If an egg is laid but not fertilized, it produces a male (or
22921:
22668:
22536:
with a Fibonacci number of nodes that has been proposed as a
22218:
is to replace 1 with 0 and 2 with 10, and drop the last zero.
21175:. Determining a general formula for the Pisano periods is an
7860:
Taking the determinant of both sides of this equation yields
5482:{\displaystyle {\vec {F}}_{n}=\mathbf {A} ^{n}{\vec {F}}_{0}}
3464:
28774:
22977:
Fibonacci numbers also appear in the ancestral pedigrees of
21152:
If the members of the Fibonacci sequence are taken mod
17980:, any Fibonacci prime must have a prime index. As there are
17683:{\displaystyle n\mid F_{n\;-\,\left({\frac {5}{n}}\right)},}
17613:{\displaystyle p\mid F_{p\;-\,\left({\frac {5}{p}}\right)}.}
16130:
The sum of all even-indexed reciprocal Fibonacci numbers is
8681:, meaning no such sequence exists whose sum is â1, and
5410:{\displaystyle {\vec {F}}_{k+1}=\mathbf {A} {\vec {F}}_{k},}
186:
and with the Fibonacci numbers form a complementary pair of
26174:
25127:
24934:
24559:
24327:
24033:
24008:
23438:
22664:
21232:
Generalizing the index to negative integers to produce the
19299:
18303:
17537:
16754:
which follows from the closed form for its partial sums as
10865:
4653:. This is because Binet's formula, which can be written as
2365:
26037:, Springer Monographs in Mathematics, New York: Springer,
25942:
Ball, Keith M (2003), "8: Fibonacci's Rabbits Revisited",
24453:
Williams, H. C. (1982), "A note on the Fibonacci quotient
18034:. More generally, no Fibonacci number other than 1 can be
16888:
Every third number of the sequence is even (a multiple of
10981:{\displaystyle \operatorname {Seq} ({\mathcal {Z+Z^{2}}})}
2874:
In fact, the rounding error quickly becomes very small as
26148:
25503:
Brousseau, A (1969), "Fibonacci Statistics in Conifers",
24844:
24239:
23768:
22978:
22598:
Fibonacci numbers are used in a polyphase version of the
17723:
15530:
Fibonacci numbers can sometimes be evaluated in terms of
12926:{\displaystyle F_{m}F_{n+1}-F_{m+1}F_{n}=(-1)^{n}F_{m-n}}
11853:{\displaystyle \sum _{i=1}^{n+1}F_{i}^{2}=F_{n+1}F_{n+2}}
7671:-st convergent can be found from the recurrence relation
6840:
th element in the Fibonacci series is therefore given by
4569:
Binet's formula provides a proof that a positive integer
3662:
This convergence holds regardless of the starting values
3562:
observed that the ratio of consecutive Fibonacci numbers
178:
increases. Fibonacci numbers are also closely related to
25688:
24700:
Acta Mathematica Academiae Paedagogicae NyĂregyhĂĄziensis
23893:, The Mathematical Association of America, p. 153,
23840:"The Fibonacci Numbers and Golden section in Nature â 1"
23725:
23581:
22996:
If, however, an egg is fertilized, it produces a female.
21539:
This can be proved by expanding the generating function
18038:, and no ratio of two Fibonacci numbers can be perfect.
11923:{\displaystyle {\sqrt {5}}F_{n}=\varphi ^{n}-\psi ^{n}.}
25967:
The Art of Proof: Basic Training for Deeper Mathematics
25567:
23985:
23644:
23642:
23557:
21990:
ways to do this (equivalently, it's also the number of
15675:
and the sum of squared reciprocal Fibonacci numbers as
7577:
7565:
7551:
7539:
7525:
7513:
7434:
7390:
6572:
Equivalently, the same computation may be performed by
6533:
6511:
6488:
6476:
6445:
6423:
6400:
6388:
6277:
6255:
6232:
6220:
6166:
6144:
6121:
6109:
5185:
In particular, the left-hand side is a perfect square.
3934:
this expression can be used to decompose higher powers
2480:
to be arbitrary constants, a more general solution is:
25795:
25122:
24929:
24664:
Cohn, J. H. E. (1964), "On square Fibonacci numbers",
24560:"Sequence A005478 (Prime Fibonacci numbers)"
24554:
24322:
24028:
24003:
23519:
23517:
23433:
23361: â Organization for research on Fibonacci numbers
22916:
is a constant scaling factor; the florets thus lie on
21856:
20636:
20609:
20569:
20542:
20510:
20251:
20224:
20181:
20154:
20122:
19866:
19839:
19799:
19772:
19740:
19475:
19448:
19405:
19378:
19346:
19216:
19189:
19107:
19080:
18827:
18731:
18663:
18567:
18471:
18127:
17836:
17756:
16349:
16259:
15254:
15197:
15139:
15131:
14402:
12548:
12516:
11939:
11168:
Fibonacci identities often can be easily proved using
10533:
In words, the sum of the first Fibonacci numbers with
9857:
This may be seen by dividing all sequences summing to
7752:
7707:
7580:
7568:
7554:
7542:
7528:
7516:
7437:
7393:
7282:
7208:
7159:
7038:
6777:
6717:
6536:
6514:
6491:
6479:
6448:
6426:
6403:
6391:
6280:
6258:
6235:
6223:
6169:
6147:
6124:
6112:
5601:
5516:
5262:
2687:
1607:
28643:
1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⯠(inverses of primes)
28633:
1 â 1 + 2 â 6 + 24 â 120 + ⯠(alternating factorials)
27159:
24716:
24459:
23924:(2nd, illustrated ed.). CRC Press. p. 260.
23917:
23488:
23486:
23208:
23171:
23134:
23097:
23062:
22842:
22389:
22000:
21855:
21759:
21545:
21432:
20896:
20739:
20686:
20501:
20354:
20301:
20113:
19969:
19916:
19731:
19581:
19528:
19337:
19032:
18936:
18458:
18329:
18161:
18118:
17750:
17637:
17567:
17368:
17135:
17099:
17066:
16942:
16894:
16764:
16670:
16496:
16348:
16258:
16136:
15995:
15895:
15681:
15540:
15458:
15429:
15409:
15394:{\displaystyle s(z)=s\!\left(-{\frac {1}{z}}\right).}
15345:
15311:
15189:
15032:
14999:
14972:
14447:
14334:
14285:
14265:
14105:
13924:
13769:
13545:
13419:
13293:
13128:
12938:
12829:
12705:
12692:{\displaystyle {F_{n}}^{2}-F_{n+1}F_{n-1}=(-1)^{n-1}}
12606:
12589:
12036:
12012:
11873:
11772:
11634:
11621:{\displaystyle \sum _{i=1}^{n}F_{i}^{2}=F_{n}F_{n+1}}
11552:
11510:
11429:
11404:
11290:
11254:
11181:
11116:
11055:
11022:
10994:
10945:
10889:
10799:
10753:
10714:
10701:{\displaystyle \sum _{i=1}^{n}F_{i}^{2}=F_{n}F_{n+1}}
10632:
10591:
10543:
10460:
10381:
10305:
10272:
10082:
9972:
9889:
9863:
9788:
9746:
9694:
9661:
9628:
9602:
9576:
9397:
9370:
9305:
9133:
9023:
8943:
8881:
8822:
8769:
8720:
8687:
8654:
8627:
8601:
8574:
8226:
8009:
7870:
7700:
7497:
7373:
6848:
6705:
6594:
6371:
5901:
5777:
5662:
5574:
5508:
5425:
5351:
5205:
5072:
4938:
4823:
4790:
4752:
4659:
4619:
4583:
4500:
4415:
4376:
4339:
4075:
3995:
3967:
3940:
3896:
3856:
3780:
3722:
3695:
3668:
3594:
3572:
3487:
3409:
3359:
3292:
3251:
3122:
2992:
2902:
2806:
2766:
2551:
2488:
2388:
2304:
1938:
1875:
1724:
1646:
1460:
1419:
1325:
1251:
1137:
436:
387:
354:
279:
223:
27544:
25919:
25907:
25833:
IEEE Transactions on Instrumentation and Measurement
25670:
25555:
25440:"Phyllotaxis as a Dynamical Self Organizing Process"
25340:
23639:
7953:{\displaystyle (-1)^{n}=F_{n+1}F_{n-1}-{F_{n}}^{2}.}
3442:{\displaystyle n\log _{10}\varphi \approx 0.2090\,n}
3327:{\displaystyle \log _{10}(\varphi )=0.208987\ldots }
155:'s bracts, though they do not occur in all species.
25869:
23973:
23514:
21195:, the Pisano period may be found as an instance of
20727:{\displaystyle F_{14}=377{\text{ and }}F_{15}=610.}
17690:where the Legendre symbol has been replaced by the
4047:{\displaystyle \varphi ^{n}=F_{n}\varphi +F_{n-1}.}
1038:) plus the number of pairs alive last month (month
158:Fibonacci numbers are also strongly related to the
25863:
25622:
25179:
24640:Honsberger, Ross (1985), "Mathematical Gems III",
24486:
24355:Comptes Rendus de l'Académie des Sciences, Série I
24166:
23947:
23816:Divine Proportion: Phi In Art, Nature, and Science
23593:
23483:
23227:
23190:
23153:
23116:
23081:
22898:
22649:A one-dimensional optimization method, called the
22516:
22012:
21932:
21772:
21745:
21531:
21350:Not adding the immediately preceding numbers. The
21107:
20851:
20726:
20673:
20469:
20341:
20288:
20081:
19956:
19903:
19699:
19568:
19515:
19305:
19002:
18914:
18442:
18309:
18147:
17894:
17682:
17612:
17543:
17270:
17118:
17085:
17037:
16913:
16868:
16746:
16640:
16476:
16334:
16241:
16120:
15971:
15879:
15665:
15511:
15444:
15415:
15393:
15326:
15293:
15233:
15175:
15117:
15011:
14985:
14956:
14433:
14386:
14318:
14271:
14246:
14061:
13906:
13738:
13532:
13406:
13277:
13091:
12925:
12809:
12691:
12570:
12534:
12502:
12022:
11998:
11922:
11852:
11759:
11620:
11538:
11494:
11416:
11387:
11273:
11240:
11153:
11099:
11034:
11000:
10980:
10923:
10856:
10785:
10727:
10700:
10607:
10565:
10525:
10446:
10364:
10291:
10255:
10062:
9958:
9875:
9847:
9759:
9732:
9680:
9647:
9614:
9588:
9562:
9383:
9356:
9288:
9118:
9008:
8928:
8866:
8807:
8744:
8706:
8673:
8640:
8613:
8587:
8518:
8200:
7952:
7828:
7611:
7454:
7357:
6826:
6687:
6562:
6347:
5881:
5761:
5645:
5560:
5481:
5409:
5337:
5174:
5055:
4922:
4803:
4772:
4738:
4641:
4605:
4554:
4466:
4401:
4362:
4325:
4046:
3973:
3953:
3924:
3862:
3842:
3759:
3708:
3681:
3652:
3581:
3544:
3441:
3384:
3326:
3278:
3237:
3108:
2968:
2864:
2788:
2730:
2666:
2533:
2449:
2370:
2239:
1920:
1849:
1700:
1632:
1583:
1444:
1403:
1295:
1233:
487:
422:
373:
330:
265:
67:0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ....
26543:
26085:(in French), vol. 1, Paris: Gauthier-Villars
26061:(First trade paperback ed.), New York City:
24875:Luca, Florian; MejĂa Huguet, V. Janitzio (2010),
24847:"There are no multiply-perfect Fibonacci numbers"
24426:Su, Francis E (2000), "Fibonacci GCD's, please",
24208:
23569:
23372: â Binary sequence from Fibonacci recurrence
22656:The Fibonacci number series is used for optional
22578:The keys in the left spine are Fibonacci numbers.
22267:without an odd number of consecutive integers is
21520:
21493:
18148:{\displaystyle {\bigl (}{\tfrac {p}{5}}{\bigr )}}
18105:The divisibility of Fibonacci numbers by a prime
16932:. Thus the Fibonacci sequence is an example of a
16469:
15817:
15758:
15742:
15614:
15364:
13984:
13971:
13829:
13816:
10935:. More precisely, this sequence corresponds to a
10526:{\displaystyle \sum _{i=1}^{n}F_{2i}=F_{2n+1}-1.}
10063:{\displaystyle \{(1,1,...,1,2)\},\{(1,1,...,1)\}}
9782:-th Fibonacci number minus 1. In symbols:
9119:{\displaystyle F_{4}=3=|\{(1,1,1),(1,2),(2,1)\}|}
7342:
7329:
7138:
7111:
7014:
6987:
6942:
6915:
6889:
6856:
6753:
6551:
6463:
6332:
6304:
6295:
6206:
6193:
6184:
5816:
5803:
5750:
5722:
5694:
5681:
5635:
5550:
5329:
5296:
5248:
5209:
3554:
2789:{\displaystyle {\frac {\varphi ^{n}}{\sqrt {5}}}}
1716:satisfy the Fibonacci recursion. In other words,
981:The Fibonacci sequence first appears in the book
28904:
25731:"The Fibonacci sequence as it appears in nature"
25623:Prusinkiewicz, Przemyslaw; Hanan, James (1989),
25479:Jones, Judy; Wilson, William (2006), "Science",
25414:
24874:
18001:is one greater or one less than a prime number.
17218:
17177:
17136:
17002:
16943:
11930:This can be used to prove Fibonacci identities.
11495:{\displaystyle \sum _{i=1}^{n+1}F_{i}=F_{n+3}-1}
10447:{\displaystyle \sum _{i=0}^{n-1}F_{2i+1}=F_{2n}}
7481:This property can be understood in terms of the
3782:
3596:
922:also expresses knowledge of the sequence in the
27:Numbers obtained by adding the two previous ones
26429:
25872:Journal of Optimization Theory and Applications
25202:, Cambridge Univ. Press, p. 121, Ex 1.35,
23788:"Fibonacci's Liber Abaci (Book of Calculation)"
23459:
23457:
23455:
22642:Fibonacci numbers arise in the analysis of the
22349:The Fibonacci numbers are also an example of a
19012:Such primes (if there are any) would be called
17553:These cases can be combined into a single, non-
11241:{\displaystyle \sum _{i=1}^{n}F_{i}=F_{n+2}-1.}
11100:{\displaystyle \sum _{i=0}^{\infty }F_{i}z^{i}}
3456:there are either 4 or 5 Fibonacci numbers with
1085:
26223:
26209:
26175:sequence A000045 (Fibonacci numbers)
26005:
25991:(3rd ed.), New Jersey: World Scientific,
25829:
25589:Formaliser le vivant - Lois, Théories, ModÚles
25304:, and Appendix B, The Ring Lemma, pp. 318â321.
24666:The Journal of the London Mathematical Society
24265:Glaister, P (1995), "Fibonacci power series",
24209:Flajolet, Philippe; Sedgewick, Robert (2009),
24184:
23024:, plus the number of male ancestors, which is
23012:, is the number of female ancestors, which is
20342:{\displaystyle F_{6}=8{\text{ and }}F_{7}=13.}
17942:2, 3, 5, 13, 89, 233, 1597, 28657, 514229, ...
17726:number – then we can calculate
10864:; from this the identity follows by comparing
10365:{\displaystyle \sum _{i=1}^{n}F_{i}=F_{n+2}-1}
9848:{\displaystyle \sum _{i=1}^{n}F_{i}=F_{n+2}-1}
3890:Since the golden ratio satisfies the equation
2534:{\displaystyle U_{n}=a\varphi ^{n}+b\psi ^{n}}
1921:{\displaystyle U_{n}=a\varphi ^{n}+b\psi ^{n}}
202:The Fibonacci spiral: an approximation of the
83:The Fibonacci numbers were first described in
28790:
28279:
28129:
26195:
25964:
25537:
24598:, Cambridge University Press, pp. 1â12,
24351:
23587:
22184:without consecutive integers, that is, those
21923:
21910:
21898:
21885:
21873:
21860:
20649:
20630:
20582:
20563:
20523:
20504:
20264:
20245:
20194:
20175:
20135:
20116:
19957:{\displaystyle F_{5}=5{\text{ and }}F_{6}=8.}
19879:
19860:
19812:
19793:
19753:
19734:
19569:{\displaystyle F_{3}=2{\text{ and }}F_{4}=3.}
19488:
19469:
19418:
19399:
19359:
19340:
19229:
19210:
19120:
19101:
18924:It is not known whether there exists a prime
18840:
18821:
18744:
18725:
18676:
18657:
18580:
18561:
18484:
18465:
18140:
18121:
5638:
5614:
5553:
5529:
4467:{\displaystyle \psi ^{n}=F_{n}\psi +F_{n-1}.}
3279:{\displaystyle \ln(\varphi )=0.481211\ldots }
381:is omitted, so that the sequence starts with
28726:Hypergeometric function of a matrix argument
28031:
26381:
25437:
24797:Rendiconti del Circolo Matematico di Palermo
24533:. Williams calls this property "well known".
24235:
24233:
24231:
23875:Faculty of Engineering and Physical Sciences
23666:(Hn.). Varanasi-I: TheChowkhamba Vidyabhawan
23621:, vol. 1, Addison Wesley, p. 100,
23452:
21135:are collected at the relevant repositories.
16925:-th number of the sequence is a multiple of
14387:{\displaystyle s(z)={\frac {z}{1-z-z^{2}}}.}
10266:... where the last two terms have the value
10245:
10209:
10193:
10151:
10057:
10021:
10015:
9973:
9552:
9492:
9476:
9416:
9278:
9158:
9108:
9048:
8998:
8968:
8918:
8906:
8856:
8847:
8797:
8794:
7629:are ratios of successive Fibonacci numbers:
1107:linear recurrence with constant coefficients
1010:: how many pairs will there be in one year?
214:The Fibonacci numbers may be defined by the
28582:1 + 1/2 + 1/3 + ... (Riemann zeta function)
26032:
25478:
25369:
25344:Proceedings of the USSR Academy of Sciences
25005:
24993:
24981:
23463:
21418:The Fibonacci numbers occur as the sums of
12006:note that the left hand side multiplied by
11999:{\textstyle \sum _{i=1}^{n}F_{i}=F_{n+2}-1}
10924:{\displaystyle (F_{n})_{n\in \mathbb {N} }}
8554:
982:
964:
100:
28797:
28783:
28286:
28272:
28136:
28122:
26496:Possessing a specific set of other numbers
26319:
26202:
26188:
26035:Reciprocity Laws: From Euler to Eisenstein
25274:
25088:
24868:
24838:
24639:
24594:; Cooper, Joshua; Hurlbert, Glenn (eds.),
24379:
24215:, Cambridge University Press, p. 42,
22624:whose child trees (recursively) differ in
20798:
20797:
20791:
20790:
20416:
20415:
20409:
20408:
20028:
20027:
20021:
20020:
19643:
19642:
19636:
19635:
17698:is a prime, and if it fails to hold, then
17652:
17582:
16883:
16878:
15603:
11008:-th Fibonacci number equals the number of
10871:
7474:of â1, and thus it is a 2 Ă 2
3885:
2676:
1075:th month, the number of pairs is equal to
28638:1 + 1/2 + 1/3 + 1/4 + ⯠(harmonic series)
27959:
26906:
25891:
25766:
25502:
25300:; see especially Lemma 8.2 (Ring Lemma),
25221:
25134:On-Line Encyclopedia of Integer Sequences
24954:
24941:On-Line Encyclopedia of Integer Sequences
24730:
24566:On-Line Encyclopedia of Integer Sequences
24519:
24509:
24407:
24338:On-Line Encyclopedia of Integer Sequences
24228:
24040:On-Line Encyclopedia of Integer Sequences
24015:On-Line Encyclopedia of Integer Sequences
23702:
23506:
23470:, Indiana University Press, p. 126,
23445:On-Line Encyclopedia of Integer Sequences
22947:, the nearest neighbors of floret number
21940:, where we are choosing the positions of
21167:. The lengths of the periods for various
17742:efficiently using the matrix form. Thus
17656:
17586:
17034:
15294:{\textstyle z\mapsto -s\left(-1/z\right)}
10915:
8621:. This can be taken as the definition of
6472:
6216:
6212:
5197:that describes the Fibonacci sequence is
5168:
3435:
1391:
1286:
1096:
28293:
25965:Beck, Matthias; Geoghegan, Ross (2010),
25538:Scott, T.C.; Marketos, P. (March 2014),
24788:
24579:
24452:
24264:
24150:
23813:
23716:
23660:
23393:
23241:appears on all lines of the genealogy.)
23038:
22800:
22720:
22567:
22551:, used to prove connections between the
21957:
21405:
21243:using a modification of Binet's formula.
13746:These can be found experimentally using
12699:Catalan's identity is a generalization:
10939:. The specification of this sequence is
3925:{\displaystyle \varphi ^{2}=\varphi +1,}
3874:
3768:
3470:representation, the number of digits in
3385:{\displaystyle \varphi ^{n}/{\sqrt {5}}}
1062:
954:
804:
348:Under some older definitions, the value
197:
163:
70:
28852:Greedy algorithm for Egyptian fractions
25585:
25549:MacTutor History of Mathematics archive
25541:On the Origin of the Fibonacci Sequence
25197:
24058:, Oxford University Press, p. 92,
23885:
23605:
22583:The Fibonacci numbers are important in
22286:The number of binary strings of length
22221:The number of binary strings of length
22122:The number of binary strings of length
21974:as an ordered sum of 1s and 2s (called
21962:Use of the Fibonacci sequence to count
21138:
21117:All known factors of Fibonacci numbers
17326:is congruent to 2 or 3 modulo 5, then,
13118:. The last is an identity for doubling
12571:{\textstyle \varphi -\psi ={\sqrt {5}}}
11862:
11154:{\displaystyle {\frac {z}{1-z-z^{2}}}.}
10857:{\displaystyle F_{n},F_{n-1},...,F_{1}}
10626:A different trick may be used to prove
9299:In this manner the recurrence relation
9009:{\displaystyle F_{3}=2=|\{(1,1),(2)\}|}
8559:
4773:{\displaystyle {\sqrt {5}}\varphi ^{n}}
14:
28905:
28067:
26090:
25762:
25760:
25147:
25055:Factors of Fibonacci and Lucas numbers
24091:
24053:
23818:, New York: Sterling, pp. 20â21,
23774:
23746:"A History of Piáč
gala's Combinatorics"
23719:'VáčttajÄtisamuccaya' of kavi Virahanka
23563:
23429:
23427:
22912:is the index number of the floret and
22321:without an even number of consecutive
22290:without an even number of consecutive
17306:is congruent to 1 or 4 modulo 5, then
16650:Moreover, this number has been proved
14326:and its sum has a simple closed form:
14085:
12819:
10793:and decompose it into squares of size
9959:{\displaystyle (2,...),(1,2,...),...,}
4555:{\displaystyle F_{n}=F_{n+2}-F_{n+1}.}
3985:yield Fibonacci numbers as the linear
3449:. As a consequence, for every integer
2800:, using the nearest integer function:
1701:{\displaystyle x^{n}=x^{n-1}+x^{n-2},}
1452:, this formula can also be written as
827:) end with a short syllable and five (
266:{\displaystyle F_{0}=0,\quad F_{1}=1,}
28778:
28267:
28117:
28066:
28030:
27994:
27958:
27918:
27543:
27432:
27158:
27073:
27028:
26905:
26595:
26542:
26494:
26428:
26380:
26318:
26222:
26183:
26078:
26051:
25925:
25913:
25801:
25676:
25650:
25573:
25561:
25313:
25248:
25200:Enumerative Combinatorics I (2nd ed.)
25185:
24697:
24642:AMS Dolciani Mathematical Expositions
24240:
24172:
23837:
23731:
23682:
23648:
23611:
23575:
23523:
23492:
23273:-th Fibonacci number. (However, when
22985:), according to the following rules:
22607:. A tape-drive implementation of the
22283:is to replace 1 with 0 and 2 with 11.
22251:without an odd number of consecutive
22225:without an odd number of consecutive
21081:
20999:
18046:With the exceptions of 1, 8 and 144 (
14396:This can be proved by multiplying by
9357:{\displaystyle F_{n}=F_{n-1}+F_{n-2}}
3760:{\displaystyle U_{1}=-U_{0}/\varphi }
488:{\displaystyle F_{n}=F_{n-1}+F_{n-2}}
331:{\displaystyle F_{n}=F_{n-1}+F_{n-2}}
28878:Generalizations of Fibonacci numbers
28873:List of things named after Fibonacci
28868:Fibonacci numbers in popular culture
26596:
26128:Periods of Fibonacci Sequences Mod m
25983:
25941:
24960:The New Book of Prime Number Records
24881:Annales Mathematicae at Informaticae
24794:
24663:
24487:{\displaystyle F_{p-\varepsilon }/p}
23991:
23979:
23743:
23599:
23365:Fibonacci numbers in popular culture
21209:Generalizations of Fibonacci numbers
17722:is large – say a 500-
17622:
13122:; other identities of this type are
9740:sequences, showing this is equal to
7485:representation for the golden ratio
4476:These expressions are also true for
1445:{\displaystyle \psi =-\varphi ^{-1}}
895:) is obtained by adding one to the
91:on enumerating possible patterns of
28603:1 â 1 + 1 â 1 + ⯠(Grandi's series)
27995:
25757:
24768:
23424:
22635:Fibonacci numbers are used by some
22563:
22020:rectangle). For example, there are
21703:
20841:
20782:
20459:
20400:
20071:
20012:
19689:
19627:
19285:
19278:
19253:
19246:
19176:
19169:
19144:
19137:
18982:
18429:
18369:
18289:
18282:
18244:
18237:
17919:
17881:
17627:The above formula can be used as a
17496:
17489:
17434:
17427:
15301:is the generating function for the
13750:, and are useful in setting up the
12581:
11163:
10786:{\displaystyle F_{n}\times F_{n+1}}
10623:-th Fibonacci number minus 1.
9622:and the cardinality of each set is
6836:The closed-form expression for the
1601:are both solutions of the equation
1117:, named after French mathematician
141:the arrangement of leaves on a stem
24:
28143:
27919:
25767:Hutchison, Luke (September 2004),
25592:(in French), Hermann, p. 28,
25020:Fibonacci and Lucas factorizations
24962:, New York: Springer, p. 64,
24425:
24056:A New Year Gift: On Hexagonal Snow
22805:Illustration of Vogel's model for
21914:
21889:
21864:
21718:
21497:
21202:
16687:
16604:
16554:
16513:
16291:
16153:
16012:
15912:
15698:
15557:
15521:
14865:
14760:
14710:
14666:
14609:
14559:
14515:
14319:{\displaystyle |z|<1/\varphi ,}
14256:This series is convergent for any
14137:
13975:
13820:
12590:Cassini's and Catalan's identities
11072:
10968:
10964:
10957:
10878:
9364:may be understood by dividing the
7333:
7115:
6991:
6962:
6919:
6860:
6706:
6656:
6612:
6308:
6197:
5807:
5726:
5685:
5300:
5213:
3792:
3606:
3017:
3014:
3011:
3008:
3005:
3002:
2999:
840:The Fibonacci sequence appears in
25:
28929:
28721:Generalized hypergeometric series
26109:
25527:, Computer Science For Fun: CS4FN
25521:"Marks for the da Vinci Code: Bâ"
25483:, Ballantine Books, p. 544,
25386:Amiga ROM Kernel Reference Manual
25224:"Review of Yuri V. Matiyasevich,
25091:The American Mathematical Monthly
24912:
24771:"On triangular Fibonacci numbers"
24585:"Probabilizing Fibonacci numbers"
23864:
18041:
17991:No Fibonacci number greater than
14094:of the Fibonacci sequence is the
8929:{\displaystyle F_{2}=1=|\{(1)\}|}
7853:arithmetic operations, using the
7842:, this matrix can be computed in
5193:A 2-dimensional system of linear
4564:
4402:{\displaystyle \psi ^{2}=\psi +1}
4363:{\displaystyle \psi =-1/\varphi }
2296:satisfy the system of equations:
1121:, though it was already known by
170:-th Fibonacci number in terms of
28759:
28758:
28731:Lauricella hypergeometric series
28449:
28252:
28097:
27705:Perfect digit-to-digit invariant
27074:
25703:The Algorithmic Beauty of Plants
25154:-step Fibonacci sequence modulo
23838:Knott, Ron (25 September 2016),
22924:, approximately 137.51°, is the
22547:Fibonacci numbers appear in the
21343:, the result is the sequence of
17710:and satisfies the formula, then
14079:
8867:{\displaystyle F_{1}=1=|\{()\}|}
5653:corresponding to the respective
5450:
5381:
2796:. Therefore, it can be found by
2459:producing the required formula.
28741:Riemann's differential equation
25823:
25807:
25748:
25723:
25682:
25644:
25616:
25579:
25531:
25513:
25496:
25472:
25431:
25415:Dean Leffingwell (2021-07-01),
25408:
25392:
25378:
25363:
25334:
25319:The Art of Computer Programming
25307:
25268:
25242:
25215:
25191:
25141:
25116:
25082:
25046:
25011:
24948:
24923:
24906:
24762:
24710:
24691:
24657:
24633:
24573:
24548:
24536:
24446:
24419:
24401:
24373:
24345:
24316:
24300:
24258:
24202:
24178:
24144:
24120:
24085:
24071:
24047:
24022:
23997:
23941:
23921:Discrete Mathematics with Ducks
23911:
23879:
23858:
23831:
23807:
23780:
23737:
23710:
23676:
23654:
23618:The Art of Computer Programming
23530:The Art of Computer Programming
23400:
22734:arrangement of leaves on a stem
22614:The Art of Computer Programming
22585:computational run-time analysis
22279:. A bijection with the sums to
21396:
21217:, and specifically by a linear
20834:
20775:
20607:
20540:
20452:
20393:
20222:
20152:
20064:
20005:
19837:
19770:
19682:
19620:
19446:
19376:
18975:
18422:
18383:
18377:
18362:
18155:which is evaluated as follows:
17874:
12600:Cassini's identity states that
11398:and so we have the formula for
10988:. Indeed, as stated above, the
10937:specifiable combinatorial class
9733:{\displaystyle F_{n-1}+F_{n-2}}
6765:
5703:
3582:{\displaystyle \varphi \colon }
2878:grows, being less than 0.1 for
2431:
1930:satisfies the same recurrence,
1859:It follows that for any values
1109:, the Fibonacci numbers have a
973:Biblioteca Nazionale di Firenze
505:The first 20 Fibonacci numbers
243:
25935:
25447:Journal of Theoretical Biology
25281:, Cambridge University Press,
25060:collects all known factors of
25025:collects all known factors of
24984:, pp. 73â74, ex. 2.25â28.
24497:Canadian Mathematical Bulletin
23918:Sarah-Marie Belcastro (2018).
23464:Goonatilake, Susantha (1998),
22637:pseudorandom number generators
22471:
22432:
22420:
22390:
21684:
21671:
21637:
21624:
21608:
21596:
21422:in the "shallow" diagonals of
21401:
21246:Starting with other integers.
21191:. However, for any particular
20845:
20835:
20786:
20776:
20463:
20453:
20404:
20394:
20075:
20065:
20016:
20006:
19693:
19683:
19631:
19621:
19289:
19279:
19257:
19247:
19180:
19170:
19148:
19138:
18993:
18976:
18433:
18423:
18373:
18363:
18293:
18283:
18248:
18238:
18015:1, 3, 21, and 55 are the only
17934:is a Fibonacci number that is
17885:
17875:
17702:is definitely not a prime. If
17506:
17500:
17490:
17444:
17438:
17428:
17388:
17259:
17221:
17212:
17180:
17171:
17139:
17029:
17005:
16991:
16946:
16269:
16263:
16231:
16218:
16209:
16196:
16030:
16020:
15468:
15462:
15439:
15433:
15355:
15349:
15321:
15315:
15258:
15042:
15036:
15024:partial fraction decomposition
14921:
14870:
14489:
14483:
14477:
14452:
14428:
14403:
14344:
14338:
14295:
14287:
14115:
14109:
13256:
13246:
12898:
12888:
12773:
12763:
12674:
12664:
12596:Cassini and Catalan identities
12493:
12468:
12448:
12436:
10975:
10952:
10904:
10890:
10249:
10242:
10212:
10205:
10197:
10190:
10154:
10147:
10054:
10024:
10012:
9976:
9938:
9914:
9908:
9890:
9556:
9537:
9519:
9513:
9495:
9488:
9480:
9461:
9443:
9437:
9419:
9412:
9282:
9275:
9263:
9257:
9239:
9233:
9215:
9209:
9191:
9185:
9161:
9154:
9112:
9105:
9093:
9087:
9075:
9069:
9051:
9044:
9002:
8995:
8989:
8983:
8971:
8964:
8922:
8915:
8909:
8902:
8860:
8853:
8850:
8843:
8808:{\displaystyle F_{0}=0=|\{\}|}
8801:
8790:
8738:
8722:
8496:
8461:
8438:
8403:
8380:
8342:
7881:
7871:
7625:of the continued fraction for
7418:
7408:
7245:
7235:
7075:
7065:
6090:
6072:
6062:
6041:
5997:
5960:
5913:
5870:
5843:
5785:
5710:
5669:
5467:
5433:
5392:
5359:
5188:
5159:
5123:
5111:
5101:
5042:
4987:
4900:
4890:
4721:
4699:
4689:
4673:
4266:
4234:
4206:
4194:
4130:
4095:
3789:
3603:
3555:Limit of consecutive quotients
3312:
3306:
3264:
3258:
3232:
3226:
3205:
3199:
3180:
3174:
3160:
3154:
3142:
3136:
3080:
3060:
3029:
3023:
2912:
2906:
2087:
2049:
2040:
2002:
1549:
1539:
1500:
1490:
423:{\displaystyle F_{1}=F_{2}=1,}
139:, such as branching in trees,
87:as early as 200 BC in work by
30:For the chamber ensemble, see
13:
1:
28804:
28736:Modular hypergeometric series
28577:1/4 + 1/16 + 1/64 + 1/256 + âŻ
26544:Expressible via specific sums
25814:"Zeckendorf representation",
25438:Douady, S; Couder, Y (1996),
25222:Harizanov, Valentina (1995),
23388:
23290:for financial market trading.
22671:computers. The number series
22114:, or equivalently, among the
21753:and collecting like terms of
19026:is an odd prime number then:
18030:No Fibonacci number can be a
18019:Fibonacci numbers, which was
17694:, then this is evidence that
17342:. The remaining case is that
16921:) and, more generally, every
16488:reciprocal Fibonacci constant
12535:{\textstyle \varphi \psi =-1}
12510:as required, using the facts
11539:{\displaystyle {F_{n+1}}^{2}}
10931:is also considered using the
10299:. From this it follows that
4492:extended to negative integers
4409:and it is also the case that
3767:. This can be verified using
1067:Solution to Fibonacci rabbit
872:
193:
32:Fibonacci Sequence (ensemble)
25989:A Walk Through Combinatorics
25706:, Springer-Verlag, pp.
25665:10.1016/0025-5564(79)90080-4
25358:Soviet Mathematics - Doklady
25275:Stephenson, Kenneth (2005),
25148:LĂŒ, Kebo; Wang, Jun (2006),
24432:, et al, HMC, archived from
24310:
24189:, BirkhĂ€user, pp. 5â6,
24092:Gessel, Ira (October 1972),
24079:Strena seu de Nive Sexangula
23704:10.1016/0315-0860(85)90021-7
23508:10.1016/0315-0860(85)90021-7
23417:
22834:in 1979. This has the form
22764:form of some flowers. Field
22572:Fibonacci tree of height 6.
22383:, obtained from the formula
21156:, the resulting sequence is
20865:, all odd prime divisors of
11047:ordinary generating function
4804:{\displaystyle \varphi ^{n}}
3954:{\displaystyle \varphi ^{n}}
3337:
2277:then the resulting sequence
1119:Jacques Philippe Marie Binet
1086:Relation to the golden ratio
7:
28746:Theta hypergeometric series
27633:Multiplicative digital root
26160:Encyclopedia of Mathematics
26033:Lemmermeyer, Franz (2000),
24996:, pp. 73â74, ex. 2.28.
24749:10.4007/annals.2006.163.969
23958:10.1007/978-3-322-85165-9_6
23352:
22815:A model for the pattern of
18084:). As a result, 8 and 144 (
12578:to simplify the equations.
12023:{\displaystyle {\sqrt {5}}}
11933:For example, to prove that
11049:of the Fibonacci sequence,
4370:, it is also the case that
2462:Taking the starting values
836:) end with a long syllable.
801:Golden ratio § History
151:, and the arrangement of a
10:
28934:
28628:Infinite arithmetic series
28572:1/2 + 1/4 + 1/8 + 1/16 + âŻ
28567:1/2 â 1/4 + 1/8 â 1/16 + âŻ
27029:
26019:, Wiley, pp. 91â101,
25948:Princeton University Press
25551:, University of St Andrews
25124:Sloane, N. J. A.
24931:Sloane, N. J. A.
24556:Sloane, N. J. A.
24324:Sloane, N. J. A.
24030:Sloane, N. J. A.
24005:Sloane, N. J. A.
23435:Sloane, N. J. A.
23286:levels are widely used in
22794:, specifically as certain
22717:Golden ratio § Nature
22714:
22708:
22667:audio file format used on
22651:Fibonacci search technique
22628:by exactly 1. So it is an
22229:s is the Fibonacci number
22130:s is the Fibonacci number
21239:Generalizing the index to
21206:
21145:
17923:
15501:0.001001002003005008013021
14966:where all terms involving
13752:special number field sieve
12593:
11010:combinatorial compositions
7855:exponentiation by squaring
4483:if the Fibonacci sequence
3392:, the number of digits in
1867:, the sequence defined by
1089:
798:
789:
118:Fibonacci search technique
29:
28883:The Fibonacci Association
28860:
28839:
28812:
28754:
28711:
28655:
28590:
28559:
28552:
28522:
28491:
28484:
28458:
28447:
28370:
28314:
28305:
28250:
28151:
28093:
28076:
28062:
28040:
28026:
28004:
27990:
27968:
27954:
27927:
27914:
27890:
27844:
27804:
27755:
27729:
27710:Perfect digital invariant
27662:
27646:
27625:
27592:
27557:
27553:
27539:
27447:
27428:
27397:
27364:
27321:
27298:
27285:Superior highly composite
27175:
27171:
27154:
27082:
27069:
27037:
27024:
26912:
26901:
26863:
26854:
26832:
26789:
26751:
26742:
26675:
26617:
26608:
26604:
26591:
26549:
26538:
26501:
26490:
26438:
26424:
26387:
26376:
26329:
26314:
26232:
26218:
25884:10.1007/s10957-012-0061-2
25690:Prusinkiewicz, Przemyslaw
25198:Stanley, Richard (2011),
24380:Honsberger, Ross (1985),
24054:Kepler, Johannes (1966),
23664:PÄáčinikÄlÄ«na BhÄratavaráčŁa
23588:Beck & Geoghegan 2010
23376:Random Fibonacci sequence
23359:The Fibonacci Association
23346:Golden ratio § Music
23001:ancestors at each level,
22748:, and the family tree of
22704:
22653:, uses Fibonacci numbers.
22172:is the number of subsets
22013:{\displaystyle 2\times n}
21160:with period at most
10070:each with cardinality 1.
4494:using the Fibonacci rule
2885:, and less than 0.01 for
1113:. It has become known as
950:
143:, the fruit sprouts of a
27323:Euler's totient function
27107:EulerâJacobi pseudoprime
26382:Other polynomial numbers
25850:10.1109/tim.1985.4315428
25653:Mathematical Biosciences
25586:Varenne, Franck (2010),
25421:, Scaled Agile Framework
24678:10.1112/jlms/s1-39.1.537
24309:(1899) quoted according
24267:The Mathematical Gazette
23814:Hemenway, Priya (2005),
23244:
22788:PrzemysĆaw Prusinkiewicz
21390:n-Step Fibonacci numbers
18027:and proved by Luo Ming.
15183:is the golden ratio and
14434:{\textstyle (1-z-z^{2})}
11175:For example, reconsider
10566:{\displaystyle F_{2n-1}}
9966:until the last two sets
8555:Combinatorial identities
7663:-th convergent, and the
5771:As the initial value is
4642:{\displaystyle 5x^{2}-4}
4606:{\displaystyle 5x^{2}+4}
3983:recurrence relationships
3974:{\displaystyle \varphi }
3863:{\displaystyle \varphi }
1105:defined by a homogenous
794:
28459:Properties of sequences
27137:SomerâLucas pseudoprime
27127:LucasâCarmichael number
26962:Lazy caterer's sequence
26079:Lucas, Ădouard (1891),
25738:The Fibonacci Quarterly
25481:An Incomplete Education
25008:, p. 73, ex. 2.27.
24313:, Page 95, Exercise 3b.
24102:The Fibonacci Quarterly
24094:"Fibonacci is a Square"
23754:Northeastern University
23467:Toward a Global Science
23342:a system of composition
23228:{\displaystyle F_{5}=5}
23191:{\displaystyle F_{4}=3}
23154:{\displaystyle F_{3}=2}
23117:{\displaystyle F_{2}=1}
23082:{\displaystyle F_{1}=1}
22770:Karl Friedrich Schimper
22744:, the arrangement of a
22593:greatest common divisor
22344:Hilbert's tenth problem
21964:{1, 2}-restricted
18323:is a prime number then
17119:{\displaystyle F_{2}=1}
17086:{\displaystyle F_{1}=1}
17051:greatest common divisor
16914:{\displaystyle F_{3}=2}
16884:Divisibility properties
16879:Primes and divisibility
15012:{\displaystyle k\geq 2}
13285:by Cassini's identity.
11274:{\displaystyle F_{n+1}}
10292:{\displaystyle F_{1}=1}
9681:{\displaystyle F_{n-2}}
9648:{\displaystyle F_{n-1}}
8745:{\displaystyle |{...}|}
8707:{\displaystyle F_{1}=1}
8674:{\displaystyle F_{0}=0}
8566:combinatorial arguments
4746:, can be multiplied by
3886:Decomposition of powers
3463:More generally, in the
2677:Computation by rounding
1097:Closed-form expression
991:The Book of Calculation
374:{\displaystyle F_{0}=0}
28322:Arithmetic progression
27012:WedderburnâEtherington
26412:Lucky numbers of Euler
26091:Sigler, L. E. (2002),
25969:, New York: Springer,
25836:, IM-34 (4): 650â653,
25459:10.1006/jtbi.1996.0026
25388:, AddisonâWesley, 1991
25226:Hibert's Tenth Problem
24511:10.4153/CMB-1982-053-0
24488:
24413:My Numbers, My Friends
24332:-th Fibonacci number)"
24212:Analytic Combinatorics
24157:In honour of Fibonacci
23793:The University of Utah
23340:(1895â1943) developed
23229:
23192:
23155:
23118:
23083:
23045:
22900:
22812:
22729:
22688:Scaled Agile Framework
22620:A Fibonacci tree is a
22579:
22553:circle packing theorem
22518:
22014:
21967:
21934:
21774:
21747:
21722:
21533:
21489:
21415:
21187:or of an element in a
21109:
20853:
20728:
20675:
20471:
20343:
20290:
20083:
19958:
19905:
19701:
19570:
19517:
19307:
19004:
18916:
18444:
18311:
18149:
17910:modular exponentiation
17902:Here the matrix power
17896:
17684:
17614:
17545:
17272:
17120:
17087:
17039:
16915:
16870:
16785:
16748:
16691:
16642:
16608:
16558:
16517:
16478:
16336:
16295:
16243:
16157:
16122:
16067:
16016:
15973:
15916:
15881:
15702:
15667:
15561:
15513:
15446:
15417:
15395:
15328:
15295:
15235:
15177:
15119:
15013:
14987:
14958:
14869:
14764:
14714:
14670:
14613:
14563:
14519:
14435:
14388:
14320:
14273:
14248:
14141:
14063:
13967:
13908:
13812:
13740:
13534:
13408:
13279:
13093:
12927:
12811:
12693:
12572:
12536:
12504:
12024:
12000:
11960:
11924:
11854:
11799:
11761:
11655:
11622:
11573:
11540:
11496:
11456:
11418:
11389:
11311:
11275:
11242:
11202:
11170:mathematical induction
11155:
11101:
11076:
11036:
11002:
10982:
10925:
10858:
10787:
10735:is the product of the
10729:
10702:
10653:
10609:
10608:{\displaystyle F_{2n}}
10567:
10527:
10481:
10448:
10408:
10366:
10326:
10293:
10257:
10064:
9960:
9877:
9849:
9809:
9761:
9734:
9682:
9649:
9616:
9590:
9564:
9385:
9358:
9290:
9120:
9010:
8930:
8868:
8809:
8746:
8708:
8675:
8642:
8615:
8589:
8520:
8202:
7954:
7830:
7613:
7456:
7359:
6828:
6689:
6564:
6363:closed-form expression
6349:
5883:
5763:
5647:
5562:
5483:
5411:
5345:alternatively denoted
5339:
5176:
5057:
4924:
4805:
4774:
4740:
4643:
4607:
4573:is a Fibonacci number
4556:
4468:
4403:
4364:
4327:
4048:
3975:
3955:
3926:
3880:
3864:
3844:
3761:
3710:
3683:
3654:
3583:
3546:
3443:
3386:
3328:
3280:
3239:
3110:
2970:
2866:
2790:
2732:
2668:
2535:
2451:
2372:
2241:
1922:
1851:
1702:
1634:
1633:{\textstyle x^{2}=x+1}
1585:
1446:
1405:
1297:
1235:
1111:closed-form expression
1082:
1053:-th Fibonacci number.
1045:). The number in the
983:
978:
965:
942:
906:cases and one to the
837:
489:
424:
375:
332:
267:
211:
182:, which obey the same
147:, the flowering of an
137:in biological settings
101:
80:
28713:Hypergeometric series
28327:Geometric progression
27300:Prime omega functions
27117:Frobenius pseudoprime
26907:Combinatorial numbers
26776:Centered dodecahedral
26569:Primary pseudoperfect
25356:by Myron J. Ricci in
25251:Mathematics in School
24916:The Fibonacci numbers
24489:
24386:Mathematical Gems III
23744:Shah, Jayant (1991),
23717:Velankar, HD (1962),
23661:Agrawala, VS (1969),
23394:Explanatory footnotes
23284:Fibonacci retracement
23230:
23193:
23156:
23119:
23084:
23042:
22901:
22804:
22736:, the fruitlets of a
22724:
22709:Further information:
22571:
22519:
22015:
21961:
21935:
21775:
21773:{\displaystyle x^{n}}
21748:
21702:
21534:
21446:
21420:binomial coefficients
21409:
21345:Fibonacci polynomials
21110:
20854:
20729:
20676:
20472:
20344:
20291:
20084:
19959:
19906:
19702:
19571:
19518:
19308:
19005:
18917:
18445:
18312:
18150:
17938:. The first few are:
17897:
17716:Fibonacci pseudoprime
17685:
17631:in the sense that if
17615:
17546:
17273:
17121:
17088:
17040:
16934:divisibility sequence
16916:
16871:
16765:
16749:
16671:
16656:Richard André-Jeannin
16643:
16588:
16538:
16497:
16479:
16337:
16275:
16244:
16137:
16123:
16047:
15996:
15974:
15896:
15882:
15682:
15668:
15541:
15514:
15447:
15418:
15396:
15329:
15296:
15248:The related function
15236:
15178:
15120:
15014:
14988:
14986:{\displaystyle z^{k}}
14959:
14849:
14744:
14694:
14650:
14593:
14543:
14499:
14436:
14389:
14321:
14274:
14249:
14121:
14064:
13947:
13909:
13792:
13741:
13535:
13409:
13280:
13094:
12928:
12812:
12694:
12573:
12537:
12505:
12025:
12001:
11940:
11925:
11867:The Binet formula is
11855:
11773:
11762:
11635:
11623:
11553:
11541:
11497:
11430:
11419:
11390:
11291:
11276:
11243:
11182:
11156:
11102:
11056:
11042:using terms 1 and 2.
11037:
11003:
10983:
10926:
10859:
10788:
10730:
10728:{\displaystyle F_{n}}
10703:
10633:
10610:
10568:
10528:
10461:
10449:
10382:
10367:
10306:
10294:
10258:
10065:
9961:
9878:
9850:
9789:
9762:
9760:{\displaystyle F_{n}}
9735:
9683:
9650:
9617:
9591:
9565:
9386:
9384:{\displaystyle F_{n}}
9359:
9291:
9121:
9011:
8931:
8869:
8810:
8747:
8709:
8676:
8648:with the conventions
8643:
8641:{\displaystyle F_{n}}
8616:
8590:
8588:{\displaystyle F_{n}}
8521:
8203:
7955:
7831:
7614:
7457:
7360:
6829:
6690:
6565:
6350:
5884:
5764:
5648:
5563:
5484:
5412:
5340:
5177:
5058:
4925:
4806:
4775:
4741:
4644:
4608:
4557:
4469:
4404:
4365:
4328:
4054:This equation can be
4049:
3981:and 1. The resulting
3976:
3956:
3927:
3878:
3865:
3845:
3762:
3711:
3709:{\displaystyle U_{1}}
3684:
3682:{\displaystyle U_{0}}
3655:
3584:
3547:
3444:
3387:
3329:
3281:
3240:
3111:
2971:
2867:
2791:
2733:
2669:
2536:
2452:
2373:
2242:
1923:
1852:
1703:
1635:
1586:
1447:
1406:
1298:
1236:
1066:
958:
934:
844:, in connection with
808:
490:
425:
376:
333:
268:
201:
74:
28693:Trigonometric series
28485:Properties of series
28332:Harmonic progression
27759:-composition related
27559:Arithmetic functions
27161:Arithmetic functions
27097:Elliptic pseudoprime
26781:Centered icosahedral
26761:Centered tetrahedral
26007:Borwein, Jonathan M.
25817:Encyclopedia of Math
25694:Lindenmayer, Aristid
25360:, 3:1259â1263, 1962.
25162:Utilitas Mathematica
24457:
23873:University of Surrey
23845:University of Surrey
23686:Historia Mathematica
23495:Historia Mathematica
23206:
23169:
23132:
23095:
23060:
22840:
22796:Lindenmayer grammars
22776:discovered that the
22699:Negafibonacci coding
22609:polyphase merge sort
22387:
22362:Zeckendorf's theorem
22337:Diophantine equation
22152:without consecutive
22126:without consecutive
21998:
21853:
21757:
21543:
21430:
21181:multiplicative order
20894:
20737:
20684:
20499:
20352:
20299:
20111:
19967:
19914:
19729:
19579:
19526:
19335:
19030:
18934:
18456:
18327:
18159:
18116:
18082:Carmichael's theorem
18004:The only nontrivial
17908:is calculated using
17748:
17635:
17565:
17366:
17133:
17097:
17064:
16940:
16892:
16762:
16668:
16494:
16346:
16256:
16134:
15993:
15893:
15679:
15538:
15456:
15445:{\displaystyle s(z)}
15427:
15407:
15343:
15327:{\displaystyle s(z)}
15309:
15252:
15187:
15129:
15030:
14997:
14970:
14445:
14400:
14332:
14283:
14263:
14103:
13922:
13767:
13758:a Fibonacci number.
13543:
13417:
13291:
13126:
12936:
12827:
12703:
12604:
12546:
12514:
12034:
12010:
11937:
11871:
11863:Binet formula proofs
11770:
11632:
11550:
11508:
11427:
11402:
11288:
11281:to both sides gives
11252:
11179:
11114:
11053:
11045:It follows that the
11020:
10992:
10943:
10887:
10797:
10751:
10712:
10630:
10589:
10541:
10458:
10379:
10303:
10270:
10080:
9970:
9887:
9861:
9786:
9774:-th is equal to the
9744:
9692:
9659:
9626:
9600:
9574:
9395:
9368:
9303:
9131:
9021:
8941:
8879:
8820:
8767:
8718:
8685:
8652:
8625:
8599:
8572:
8568:using the fact that
8560:Combinatorial proofs
8330:
8224:
8210:In particular, with
8007:
7868:
7698:
7495:
7371:
6846:
6703:
6592:
6369:
5899:
5891:it follows that the
5775:
5660:
5572:
5506:
5423:
5349:
5203:
5195:difference equations
5070:
4936:
4821:
4788:
4750:
4657:
4617:
4581:
4498:
4413:
4374:
4337:
4073:
3993:
3965:
3938:
3894:
3854:
3778:
3720:
3693:
3666:
3592:
3570:
3485:
3407:
3357:
3290:
3249:
3120:
2990:
2900:
2804:
2764:
2685:
2549:
2486:
2386:
2302:
1936:
1873:
1722:
1644:
1605:
1458:
1417:
1323:
1249:
1135:
434:
385:
352:
277:
221:
37:In mathematics, the
18:Binet's formula
28889:Fibonacci Quarterly
28829:The Book of Squares
28673:Formal power series
27685:Kaprekar's constant
27205:Colossally abundant
27092:Catalan pseudoprime
26992:SchröderâHipparchus
26771:Centered octahedral
26647:Centered heptagonal
26637:Centered pentagonal
26627:Centered triangular
26227:and related numbers
26155:"Fibonacci numbers"
26082:Théorie des nombres
25842:1985ITIM...34..650P
25506:Fibonacci Quarterly
25372:Fibonacci Quarterly
25354:English translation
24741:2004math......3046B
24429:Mudd Math Fun Facts
24152:Dijkstra, Edsger W.
23994:, pp. 155â156.
23950:Der Goldene Schnitt
23937:Extract of page 260
23891:Mathematical Circus
23868:Fibonacci's Rabbits
23777:, pp. 404â405.
22970:, which depends on
22740:, the flowering of
22576:green; heights red.
21301:Primefree sequences
21219:difference equation
21215:recurrence relation
21171:form the so-called
21139:Periodicity modulo
19014:WallâSunâSun primes
17914:adapted to matrices
17557:formula, using the
17349:, and in this case
16758:tends to infinity:
16664:gives the identity
15526:Infinite sums over
15336:functional equation
14092:generating function
14086:Generating function
12820:d'Ocagne's identity
11814:
11670:
11588:
11417:{\displaystyle n+1}
11035:{\displaystyle n-1}
10668:
9876:{\displaystyle n+1}
9615:{\displaystyle n-3}
9589:{\displaystyle n-2}
8614:{\displaystyle n-1}
8321:
7579:
7567:
7553:
7541:
7527:
7515:
7436:
7392:
7367:which again yields
6582:through use of its
6535:
6513:
6490:
6478:
6447:
6425:
6402:
6390:
6279:
6257:
6234:
6222:
6168:
6146:
6123:
6111:
2380:which has solution
2257:are chosen so that
430:and the recurrence
216:recurrence relation
206:created by drawing
184:recurrence relation
113:Fibonacci Quarterly
49:, commonly denoted
28847:Fibonacci sequence
28471:Monotonic function
28390:Fibonacci sequence
28176:Fibonacci sequence
28103:Mathematics portal
28045:Aronson's sequence
27791:SmarandacheâWellin
27548:-dependent numbers
27255:Primitive abundant
27142:Strong pseudoprime
27132:Perrin pseudoprime
27112:Fermat pseudoprime
27052:Wolstenholme prime
26876:Squared triangular
26662:Centered decagonal
26657:Centered nonagonal
26652:Centered octagonal
26642:Centered hexagonal
26139:Fibonacci Sequence
25576:, pp. 112â13.
24809:10.1007/BF02904236
24769:Luo, Ming (1989),
24521:10338.dmlcz/137492
24484:
24246:"Fibonacci Number"
24242:Weisstein, Eric W.
23796:, 13 December 2009
23734:, p. 197â198.
23566:, pp. 404â05.
23338:Joseph Schillinger
23288:technical analysis
23256:refractive indexes
23239:population founder
23225:
23188:
23151:
23114:
23079:
23046:
22896:
22813:
22730:
22711:Patterns in nature
22589:Euclid's algorithm
22580:
22542:parallel computing
22514:
22381:Pythagorean triple
22010:
21968:
21930:
21929:
21770:
21743:
21529:
21416:
21105:
21085:
21003:
20849:
20724:
20671:
20645:
20618:
20578:
20551:
20519:
20467:
20339:
20286:
20260:
20233:
20190:
20163:
20131:
20079:
19954:
19901:
19875:
19848:
19808:
19781:
19749:
19697:
19566:
19513:
19484:
19457:
19414:
19387:
19355:
19303:
19298:
19225:
19198:
19116:
19089:
19000:
18912:
18910:
18836:
18740:
18672:
18576:
18480:
18440:
18307:
18302:
18145:
18136:
18109:is related to the
17892:
17861:
17821:
17680:
17610:
17541:
17536:
17268:
17116:
17083:
17035:
16911:
16866:
16744:
16638:
16474:
16473:
16332:
16331:
16239:
16118:
15969:
15877:
15663:
15509:
15442:
15413:
15391:
15324:
15291:
15231:
15206:
15173:
15148:
15115:
15009:
14983:
14954:
14952:
14431:
14384:
14316:
14269:
14244:
14059:
13904:
13736:
13530:
13404:
13275:
13089:
12923:
12807:
12689:
12568:
12532:
12500:
12498:
12020:
11996:
11920:
11850:
11800:
11757:
11656:
11618:
11574:
11536:
11492:
11414:
11385:
11271:
11238:
11151:
11097:
11032:
10998:
10978:
10921:
10854:
10783:
10725:
10698:
10654:
10605:
10563:
10523:
10444:
10362:
10289:
10253:
10060:
9956:
9873:
9845:
9757:
9730:
9688:giving a total of
9678:
9645:
9612:
9586:
9560:
9381:
9354:
9286:
9116:
9006:
8926:
8864:
8805:
8742:
8704:
8671:
8638:
8611:
8585:
8516:
8514:
8198:
8196:
7950:
7862:Cassini's identity
7826:
7817:
7732:
7609:
7602:
7597:
7592:
7574:
7548:
7522:
7483:continued fraction
7452:
7445:
7431:
7355:
7353:
7320:
7259:
7197:
7089:
6824:
6815:
6756:
6685:
6683:
6584:eigendecomposition
6560:
6542:
6530:
6499:
6485:
6454:
6442:
6411:
6397:
6345:
6343:
6286:
6274:
6243:
6229:
6175:
6163:
6132:
6118:
5879:
5759:
5643:
5610:
5558:
5525:
5479:
5407:
5335:
5287:
5172:
5063:, it follows that
5053:
4932:Comparing this to
4920:
4801:
4782:quadratic equation
4770:
4736:
4639:
4603:
4552:
4464:
4399:
4360:
4323:
4044:
3971:
3951:
3922:
3881:
3860:
3840:
3796:
3757:
3706:
3679:
3650:
3610:
3579:
3542:
3439:
3382:
3324:
3276:
3235:
3106:
2978:Instead using the
2966:
2862:
2786:
2728:
2664:
2662:
2531:
2447:
2368:
2363:
2237:
2235:
1918:
1847:
1845:
1698:
1630:
1581:
1442:
1401:
1293:
1231:
1083:
1027:At the end of the
979:
842:Indian mathematics
838:
485:
420:
371:
328:
263:
212:
85:Indian mathematics
81:
39:Fibonacci sequence
28918:Integer sequences
28913:Fibonacci numbers
28900:
28899:
28772:
28771:
28703:Generating series
28651:
28650:
28623:1 â 2 + 4 â 8 + âŻ
28618:1 + 2 + 4 + 8 + âŻ
28613:1 â 2 + 3 â 4 + âŻ
28608:1 + 2 + 3 + 4 + âŻ
28598:1 + 1 + 1 + 1 + âŻ
28548:
28547:
28476:Periodic sequence
28445:
28444:
28430:Triangular number
28420:Pentagonal number
28400:Heptagonal number
28385:Complete sequence
28307:Integer sequences
28261:
28260:
28230:Supersilver ratio
28211:Supergolden ratio
28111:
28110:
28089:
28088:
28058:
28057:
28022:
28021:
27986:
27985:
27950:
27949:
27910:
27909:
27906:
27905:
27725:
27724:
27535:
27534:
27424:
27423:
27420:
27419:
27366:Aliquot sequences
27177:Divisor functions
27150:
27149:
27122:Lucas pseudoprime
27102:Euler pseudoprime
27087:Carmichael number
27065:
27064:
27020:
27019:
26897:
26896:
26893:
26892:
26889:
26888:
26850:
26849:
26738:
26737:
26695:Square triangular
26587:
26586:
26534:
26533:
26486:
26485:
26420:
26419:
26372:
26371:
26310:
26309:
26102:978-0-387-95419-6
26044:978-3-540-66957-9
26026:978-0-471-31515-5
26011:Borwein, Peter B.
25998:978-981-4335-23-2
25976:978-1-4419-7022-0
25957:978-0-691-11321-0
25946:, Princeton, NJ:
25804:, pp. 98â99.
25717:978-0-387-97297-8
25638:978-0-387-97092-9
25490:978-0-7394-7582-9
25328:978-0-201-89683-1
25288:978-0-521-82356-2
25209:978-1-107-60262-5
25137:, OEIS Foundation
24969:978-0-387-94457-9
24944:, OEIS Foundation
24725:(163): 969â1018,
24651:978-0-88385-318-4
24605:978-1-107-15398-1
24569:, OEIS Foundation
24415:, Springer-Verlag
24382:"Millin's series"
24341:, OEIS Foundation
24196:978-3-7643-6135-8
24187:Fibonacci Numbers
24065:978-0-19-858120-8
24043:, OEIS Foundation
24018:, OEIS Foundation
23967:978-3-8154-2511-4
23931:978-1-351-68369-2
23900:978-0-88385-506-5
23628:978-81-7758-754-8
23540:978-0-321-33570-8
23477:978-0-253-33388-9
23448:, OEIS Foundation
23308:golden ratio base
23262:reflections, for
22920:. The divergence
22894:
22879:
22869:
22819:in the head of a
22658:lossy compression
22611:was described in
22591:to determine the
22351:complete sequence
22333:Yuri Matiyasevich
22211:with the sums to
22160:. Equivalently,
22158:Fibbinary numbers
22095:
22094:
21921:
21896:
21871:
21845:
21844:
21575:
21518:
21482:
21424:Pascal's triangle
21412:Pascal's triangle
21066:
21044:
21022:
20984:
20962:
20940:
20918:
20795:
20706:
20644:
20617:
20577:
20550:
20518:
20413:
20321:
20259:
20232:
20189:
20162:
20130:
20025:
19936:
19874:
19847:
19807:
19780:
19748:
19640:
19548:
19483:
19456:
19413:
19386:
19354:
19266:
19224:
19197:
19157:
19115:
19088:
19058:
18960:
18835:
18739:
18671:
18575:
18479:
18407:
18381:
18355:
18267:
18222:
18199:
18174:
18135:
17986:composite numbers
17970:, so, apart from
17669:
17623:Primality testing
17599:
16861:
16808:
16739:
16733:
16714:
16627:
16583:
16533:
16462:
16421:
16378:
16368:
16326:
16186:
16176:
16113:
16101:
16088:
15964:
15960:
15949:
15850:
15844:
15791:
15785:
15740:
15727:
15647:
15641:
15601:
15597:
15586:
15495:
15482:
15416:{\displaystyle z}
15381:
15224:
15205:
15166:
15147:
15108:
15084:
15058:
15057:
14379:
14272:{\displaystyle z}
13982:
13916:or alternatively
13827:
13748:lattice reduction
12566:
12466:
12386:
12312:
12273:
12227:
12185:
12018:
11879:
11546:to both sides of
11146:
11109:rational function
11001:{\displaystyle n}
7669: + 1)
7604:
7599:
7594:
7578:
7566:
7552:
7540:
7526:
7514:
7476:unimodular matrix
7447:
7443:
7435:
7391:
7340:
7275:
7274:
7136:
7012:
6940:
6887:
6544:
6534:
6527:
6512:
6501:
6497:
6489:
6477:
6456:
6446:
6439:
6424:
6413:
6409:
6401:
6389:
6330:
6288:
6278:
6271:
6256:
6245:
6241:
6233:
6221:
6204:
6177:
6167:
6160:
6145:
6134:
6130:
6122:
6110:
6093:
6060:
6059:
6044:
6023:
6022:
6000:
5979:
5978:
5963:
5942:
5941:
5916:
5873:
5862:
5861:
5846:
5835:
5834:
5814:
5788:
5748:
5713:
5692:
5672:
5634:
5630:
5609:
5549:
5545:
5524:
5470:
5436:
5395:
5362:
5327:
5246:
5005:
4915:
4909:
4855:
4813:quadratic formula
4758:
4734:
3825:
3781:
3639:
3595:
3537:
3481:is asymptotic to
3403:is asymptotic to
3380:
3093:
3058:
2956:
2941:
2852:
2841:
2840:
2784:
2783:
2726:
2709:
2708:
2655:
2654:
2601:
2600:
2426:
2425:
2411:
1708:so the powers of
1576:
1518:
1517:
1380:
1352:
1346:
1278:
1272:
1226:
1225:
1189:
1123:Abraham de Moivre
1049:-th month is the
785:
784:
47:Fibonacci numbers
16:(Redirected from
28925:
28799:
28792:
28785:
28776:
28775:
28762:
28761:
28688:Dirichlet series
28557:
28556:
28489:
28488:
28453:
28425:Polygonal number
28405:Hexagonal number
28378:
28312:
28311:
28288:
28281:
28274:
28265:
28264:
28256:
28138:
28131:
28124:
28115:
28114:
28101:
28064:
28063:
28033:Natural language
28028:
28027:
27992:
27991:
27960:Generated via a
27956:
27955:
27916:
27915:
27821:Digit-reassembly
27786:Self-descriptive
27590:
27589:
27555:
27554:
27541:
27540:
27492:LucasâCarmichael
27482:Harmonic divisor
27430:
27429:
27356:Sparsely totient
27331:Highly cototient
27240:Multiply perfect
27230:Highly composite
27173:
27172:
27156:
27155:
27071:
27070:
27026:
27025:
27007:Telephone number
26903:
26902:
26861:
26860:
26842:Square pyramidal
26824:Stella octangula
26749:
26748:
26615:
26614:
26606:
26605:
26598:Figurate numbers
26593:
26592:
26540:
26539:
26492:
26491:
26426:
26425:
26378:
26377:
26316:
26315:
26220:
26219:
26204:
26197:
26190:
26181:
26180:
26173:
26168:
26118:
26105:
26086:
26075:
26047:
26029:
26001:
25979:
25960:
25929:
25923:
25917:
25911:
25905:
25904:
25895:
25867:
25861:
25860:
25827:
25821:
25820:
25811:
25805:
25799:
25793:
25792:
25791:
25790:
25784:
25778:, archived from
25773:
25764:
25755:
25752:
25746:
25745:
25744:(1): 53â56, 1963
25735:
25727:
25721:
25720:
25686:
25680:
25674:
25668:
25667:
25648:
25642:
25641:
25620:
25614:
25613:
25607:
25606:
25583:
25577:
25571:
25565:
25559:
25553:
25552:
25546:
25535:
25529:
25528:
25517:
25511:
25510:
25500:
25494:
25493:
25476:
25470:
25469:
25467:
25461:, archived from
25444:
25435:
25429:
25428:
25427:
25426:
25412:
25406:
25405:
25396:
25390:
25389:
25382:
25376:
25375:
25367:
25361:
25352:
25338:
25332:
25331:
25311:
25305:
25299:
25272:
25266:
25265:
25246:
25240:
25239:
25219:
25213:
25212:
25195:
25189:
25183:
25177:
25176:
25157:
25153:
25145:
25139:
25138:
25120:
25114:
25113:
25086:
25080:
25078:
25070:
25059:
25050:
25044:
25042:
25035:
25024:
25015:
25009:
25006:Lemmermeyer 2000
25003:
24997:
24994:Lemmermeyer 2000
24991:
24985:
24982:Lemmermeyer 2000
24979:
24973:
24972:
24956:Ribenboim, Paulo
24952:
24946:
24945:
24927:
24921:
24920:
24910:
24904:
24903:
24872:
24866:
24865:
24842:
24836:
24835:
24792:
24786:
24785:
24778:Fibonacci Quart.
24775:
24766:
24760:
24759:
24734:
24714:
24708:
24707:
24695:
24689:
24688:
24661:
24655:
24654:
24637:
24631:
24630:
24629:
24628:
24622:
24616:, archived from
24589:
24577:
24571:
24570:
24552:
24546:
24540:
24534:
24532:
24523:
24513:
24493:
24491:
24490:
24485:
24480:
24475:
24474:
24450:
24444:
24443:
24442:
24441:
24423:
24417:
24416:
24409:Ribenboim, Paulo
24405:
24399:
24398:
24377:
24371:
24370:
24349:
24343:
24342:
24331:
24320:
24314:
24304:
24298:
24297:
24262:
24256:
24255:
24254:
24237:
24226:
24225:
24206:
24200:
24199:
24182:
24176:
24170:
24164:
24163:
24162:
24148:
24142:
24141:
24139:
24138:
24124:
24118:
24117:
24116:
24114:
24098:
24089:
24083:
24082:
24075:
24069:
24068:
24051:
24045:
24044:
24026:
24020:
24019:
24001:
23995:
23989:
23983:
23977:
23971:
23970:
23945:
23939:
23935:
23915:
23909:
23908:
23883:
23877:
23876:
23862:
23856:
23855:
23854:
23852:
23835:
23829:
23828:
23811:
23805:
23804:
23803:
23801:
23784:
23778:
23772:
23766:
23765:
23764:
23762:
23750:
23741:
23735:
23729:
23723:
23722:
23714:
23708:
23707:
23706:
23680:
23674:
23673:
23658:
23652:
23646:
23637:
23636:
23609:
23603:
23597:
23591:
23585:
23579:
23573:
23567:
23561:
23555:
23554:
23551:
23521:
23512:
23511:
23510:
23490:
23481:
23480:
23461:
23450:
23449:
23431:
23411:
23404:
23312:
23279:
23272:
23268:
23261:
23236:
23234:
23232:
23231:
23226:
23218:
23217:
23199:
23197:
23195:
23194:
23189:
23181:
23180:
23162:
23160:
23158:
23157:
23152:
23144:
23143:
23125:
23123:
23121:
23120:
23115:
23107:
23106:
23088:
23086:
23085:
23080:
23072:
23071:
23044:Relationships".)
23035:
23023:
23011:
22973:
22969:
22965:
22950:
22946:
22915:
22911:
22905:
22903:
22902:
22897:
22895:
22890:
22877:
22870:
22868:
22867:
22858:
22850:
22833:
22823:was proposed by
22811:
22726:Yellow chamomile
22694:Fibonacci coding
22606:
22564:Computer science
22538:network topology
22534:undirected graph
22523:
22521:
22520:
22515:
22510:
22509:
22504:
22503:
22502:
22479:
22478:
22469:
22468:
22453:
22452:
22428:
22427:
22418:
22417:
22402:
22401:
22366:Fibonacci coding
22328:
22324:
22320:
22309:
22297:
22293:
22289:
22282:
22278:
22266:
22258:
22254:
22250:
22240:
22228:
22224:
22217:
22206:
22202:
22187:
22183:
22175:
22171:
22155:
22151:
22141:
22129:
22125:
22118:of a given set.
22091:
22086:
22081:
22072:
22067:
22062:
22057:
22052:
22047:
22042:
22041:
22036:
22019:
22017:
22016:
22011:
21989:
21973:
21965:
21954:
21943:
21939:
21937:
21936:
21931:
21928:
21927:
21926:
21913:
21903:
21902:
21901:
21888:
21878:
21877:
21876:
21863:
21841:
21836:
21831:
21822:
21817:
21812:
21807:
21798:
21793:
21788:
21787:
21779:
21777:
21776:
21771:
21769:
21768:
21752:
21750:
21749:
21744:
21742:
21741:
21732:
21731:
21721:
21716:
21692:
21691:
21670:
21669:
21645:
21644:
21623:
21622:
21595:
21594:
21576:
21574:
21573:
21572:
21547:
21538:
21536:
21535:
21530:
21525:
21524:
21523:
21514:
21496:
21488:
21487:
21483:
21478:
21467:
21460:
21442:
21441:
21384:
21352:Padovan sequence
21342:
21338:
21298:
21269:
21259:
21194:
21170:
21166:
21155:
21134:
21127:
21114:
21112:
21111:
21106:
21086:
21076:
21075:
21064:
21054:
21053:
21042:
21032:
21031:
21020:
21004:
20994:
20993:
20982:
20972:
20971:
20960:
20950:
20949:
20938:
20928:
20927:
20916:
20906:
20905:
20886:
20875:
20864:
20858:
20856:
20855:
20850:
20848:
20820:
20819:
20814:
20813:
20812:
20796:
20793:
20789:
20761:
20760:
20755:
20754:
20753:
20733:
20731:
20730:
20725:
20717:
20716:
20707:
20704:
20696:
20695:
20680:
20678:
20677:
20672:
20664:
20660:
20653:
20652:
20646:
20637:
20634:
20633:
20619:
20610:
20597:
20593:
20586:
20585:
20579:
20570:
20567:
20566:
20552:
20543:
20527:
20526:
20520:
20511:
20508:
20507:
20494:
20487:
20476:
20474:
20473:
20468:
20466:
20438:
20437:
20432:
20431:
20430:
20414:
20411:
20407:
20376:
20375:
20370:
20369:
20368:
20348:
20346:
20345:
20340:
20332:
20331:
20322:
20319:
20311:
20310:
20295:
20293:
20292:
20287:
20279:
20275:
20268:
20267:
20261:
20252:
20249:
20248:
20234:
20225:
20209:
20205:
20198:
20197:
20191:
20182:
20179:
20178:
20164:
20155:
20139:
20138:
20132:
20123:
20120:
20119:
20106:
20099:
20088:
20086:
20085:
20080:
20078:
20050:
20049:
20044:
20043:
20042:
20026:
20023:
20019:
19991:
19990:
19985:
19984:
19983:
19963:
19961:
19960:
19955:
19947:
19946:
19937:
19934:
19926:
19925:
19910:
19908:
19907:
19902:
19894:
19890:
19883:
19882:
19876:
19867:
19864:
19863:
19849:
19840:
19827:
19823:
19816:
19815:
19809:
19800:
19797:
19796:
19782:
19773:
19757:
19756:
19750:
19741:
19738:
19737:
19724:
19717:
19706:
19704:
19703:
19698:
19696:
19665:
19664:
19659:
19658:
19657:
19641:
19638:
19634:
19603:
19602:
19597:
19596:
19595:
19575:
19573:
19572:
19567:
19559:
19558:
19549:
19546:
19538:
19537:
19522:
19520:
19519:
19514:
19503:
19499:
19492:
19491:
19485:
19476:
19473:
19472:
19458:
19449:
19433:
19429:
19422:
19421:
19415:
19406:
19403:
19402:
19388:
19379:
19363:
19362:
19356:
19347:
19344:
19343:
19330:
19323:
19312:
19310:
19309:
19304:
19302:
19301:
19292:
19267:
19264:
19260:
19244:
19240:
19233:
19232:
19226:
19217:
19214:
19213:
19199:
19190:
19183:
19158:
19155:
19151:
19135:
19131:
19124:
19123:
19117:
19108:
19105:
19104:
19090:
19081:
19067:
19066:
19061:
19060:
19059:
19054:
19043:
19025:
19009:
19007:
19006:
19001:
18996:
18992:
18991:
18967:
18966:
18965:
18961:
18953:
18927:
18921:
18919:
18918:
18913:
18911:
18897:
18896:
18872:
18871:
18844:
18843:
18837:
18828:
18825:
18824:
18801:
18800:
18776:
18775:
18748:
18747:
18741:
18732:
18729:
18728:
18705:
18704:
18680:
18679:
18673:
18664:
18661:
18660:
18637:
18636:
18612:
18611:
18584:
18583:
18577:
18568:
18565:
18564:
18541:
18540:
18516:
18515:
18488:
18487:
18481:
18472:
18469:
18468:
18449:
18447:
18446:
18441:
18436:
18414:
18413:
18412:
18408:
18400:
18382:
18379:
18376:
18360:
18356:
18348:
18339:
18338:
18322:
18316:
18314:
18313:
18308:
18306:
18305:
18296:
18268:
18265:
18251:
18223:
18220:
18200:
18197:
18179:
18175:
18167:
18154:
18152:
18151:
18146:
18144:
18143:
18137:
18128:
18125:
18124:
18108:
18101:
18092:
18079:
18070:
18061:
18036:multiply perfect
18000:
17982:arbitrarily long
17979:
17969:
17959:is divisible by
17958:
17920:Fibonacci primes
17907:
17901:
17899:
17898:
17893:
17888:
17872:
17871:
17866:
17865:
17826:
17825:
17818:
17817:
17800:
17799:
17786:
17785:
17774:
17773:
17741:
17721:
17713:
17705:
17701:
17697:
17689:
17687:
17686:
17681:
17676:
17675:
17674:
17670:
17662:
17619:
17617:
17616:
17611:
17606:
17605:
17604:
17600:
17592:
17550:
17548:
17547:
17542:
17540:
17539:
17530:
17529:
17503:
17468:
17467:
17441:
17406:
17405:
17352:
17348:
17341:
17329:
17325:
17321:
17309:
17305:
17298:
17294:
17284:
17277:
17275:
17274:
17269:
17258:
17257:
17239:
17238:
17211:
17210:
17192:
17191:
17170:
17169:
17151:
17150:
17125:
17123:
17122:
17117:
17109:
17108:
17092:
17090:
17089:
17084:
17076:
17075:
17048:
17044:
17042:
17041:
17036:
17033:
17032:
16984:
16983:
16971:
16970:
16958:
16957:
16924:
16920:
16918:
16917:
16912:
16904:
16903:
16875:
16873:
16872:
16867:
16862:
16860:
16859:
16858:
16857:
16843:
16842:
16835:
16834:
16820:
16809:
16807:
16806:
16805:
16804:
16787:
16784:
16779:
16757:
16753:
16751:
16750:
16745:
16740:
16735:
16734:
16729:
16720:
16715:
16713:
16712:
16711:
16710:
16693:
16690:
16685:
16647:
16645:
16644:
16639:
16628:
16626:
16625:
16610:
16607:
16602:
16584:
16582:
16581:
16560:
16557:
16552:
16534:
16532:
16531:
16519:
16516:
16511:
16483:
16481:
16480:
16475:
16468:
16464:
16463:
16461:
16460:
16459:
16440:
16439:
16427:
16422:
16420:
16419:
16418:
16399:
16398:
16386:
16379:
16374:
16369:
16367:
16366:
16351:
16341:
16339:
16338:
16333:
16327:
16325:
16324:
16323:
16307:
16306:
16297:
16294:
16289:
16248:
16246:
16245:
16240:
16238:
16234:
16230:
16229:
16208:
16207:
16187:
16182:
16177:
16175:
16174:
16159:
16156:
16151:
16127:
16125:
16124:
16119:
16114:
16109:
16102:
16097:
16094:
16089:
16087:
16086:
16085:
16080:
16079:
16078:
16066:
16061:
16045:
16044:
16043:
16018:
16015:
16010:
15978:
15976:
15975:
15970:
15965:
15956:
15955:
15950:
15948:
15947:
15946:
15918:
15915:
15910:
15886:
15884:
15883:
15878:
15873:
15869:
15862:
15861:
15856:
15852:
15851:
15846:
15845:
15840:
15831:
15816:
15815:
15803:
15802:
15797:
15793:
15792:
15787:
15786:
15781:
15772:
15757:
15756:
15741:
15733:
15728:
15726:
15725:
15720:
15719:
15718:
15704:
15701:
15696:
15672:
15670:
15669:
15664:
15659:
15658:
15653:
15649:
15648:
15643:
15642:
15637:
15628:
15613:
15612:
15602:
15593:
15592:
15587:
15585:
15584:
15563:
15560:
15555:
15518:
15516:
15515:
15510:
15496:
15488:
15483:
15475:
15452:. For example,
15451:
15449:
15448:
15443:
15422:
15420:
15419:
15414:
15400:
15398:
15397:
15392:
15387:
15383:
15382:
15374:
15333:
15331:
15330:
15325:
15300:
15298:
15297:
15292:
15290:
15286:
15282:
15240:
15238:
15237:
15232:
15230:
15226:
15225:
15220:
15207:
15198:
15182:
15180:
15179:
15174:
15172:
15168:
15167:
15162:
15149:
15140:
15124:
15122:
15121:
15116:
15114:
15110:
15109:
15107:
15090:
15085:
15083:
15066:
15059:
15053:
15049:
15018:
15016:
15015:
15010:
14992:
14990:
14989:
14984:
14982:
14981:
14963:
14961:
14960:
14955:
14953:
14937:
14933:
14932:
14920:
14919:
14901:
14900:
14882:
14881:
14868:
14863:
14845:
14844:
14829:
14828:
14813:
14812:
14794:
14790:
14789:
14780:
14779:
14763:
14758:
14740:
14739:
14730:
14729:
14713:
14708:
14690:
14689:
14680:
14679:
14669:
14664:
14643:
14639:
14638:
14623:
14622:
14612:
14607:
14589:
14588:
14573:
14572:
14562:
14557:
14539:
14538:
14529:
14528:
14518:
14513:
14476:
14475:
14440:
14438:
14437:
14432:
14427:
14426:
14393:
14391:
14390:
14385:
14380:
14378:
14377:
14376:
14351:
14325:
14323:
14322:
14317:
14309:
14298:
14290:
14278:
14276:
14275:
14270:
14253:
14251:
14250:
14245:
14234:
14233:
14218:
14217:
14202:
14201:
14186:
14185:
14161:
14160:
14151:
14150:
14140:
14135:
14077:
14068:
14066:
14065:
14060:
14055:
14054:
14043:
14042:
14041:
14024:
14023:
14018:
14017:
14016:
14005:
14004:
13989:
13988:
13987:
13974:
13966:
13961:
13943:
13942:
13913:
13911:
13910:
13905:
13900:
13899:
13888:
13887:
13886:
13869:
13868:
13863:
13862:
13861:
13850:
13849:
13834:
13833:
13832:
13819:
13811:
13806:
13788:
13787:
13761:More generally,
13745:
13743:
13742:
13737:
13735:
13731:
13730:
13729:
13724:
13723:
13722:
13699:
13698:
13693:
13692:
13691:
13675:
13674:
13669:
13668:
13667:
13650:
13646:
13645:
13644:
13639:
13638:
13637:
13620:
13619:
13614:
13613:
13612:
13590:
13589:
13574:
13573:
13558:
13557:
13539:
13537:
13536:
13531:
13529:
13528:
13523:
13522:
13521:
13507:
13506:
13497:
13496:
13491:
13490:
13489:
13466:
13465:
13460:
13459:
13458:
13438:
13437:
13413:
13411:
13410:
13405:
13403:
13402:
13397:
13396:
13395:
13381:
13380:
13375:
13374:
13373:
13362:
13361:
13340:
13339:
13334:
13333:
13332:
13312:
13311:
13284:
13282:
13281:
13276:
13274:
13273:
13264:
13263:
13239:
13238:
13233:
13232:
13231:
13214:
13213:
13198:
13197:
13182:
13181:
13166:
13165:
13160:
13159:
13158:
13141:
13140:
13121:
13113:
13109:
13098:
13096:
13095:
13090:
13088:
13087:
13078:
13077:
13065:
13061:
13060:
13059:
13041:
13040:
13020:
13019:
13007:
13006:
13001:
13000:
12999:
12979:
12978:
12973:
12972:
12971:
12951:
12950:
12932:
12930:
12929:
12924:
12922:
12921:
12906:
12905:
12884:
12883:
12874:
12873:
12855:
12854:
12839:
12838:
12816:
12814:
12813:
12808:
12806:
12805:
12800:
12799:
12798:
12787:
12786:
12759:
12758:
12743:
12742:
12724:
12723:
12718:
12717:
12716:
12698:
12696:
12695:
12690:
12688:
12687:
12660:
12659:
12644:
12643:
12625:
12624:
12619:
12618:
12617:
12582:Other identities
12577:
12575:
12574:
12569:
12567:
12562:
12541:
12539:
12538:
12533:
12509:
12507:
12506:
12501:
12499:
12486:
12485:
12467:
12462:
12454:
12432:
12431:
12413:
12412:
12391:
12387:
12385:
12377:
12370:
12369:
12345:
12344:
12325:
12317:
12313:
12311:
12303:
12296:
12295:
12279:
12274:
12272:
12264:
12257:
12256:
12240:
12232:
12228:
12226:
12215:
12208:
12207:
12191:
12186:
12184:
12173:
12166:
12165:
12149:
12141:
12137:
12133:
12132:
12131:
12113:
12112:
12083:
12082:
12064:
12063:
12029:
12027:
12026:
12021:
12019:
12014:
12005:
12003:
12002:
11997:
11989:
11988:
11970:
11969:
11959:
11954:
11929:
11927:
11926:
11921:
11916:
11915:
11903:
11902:
11890:
11889:
11880:
11875:
11859:
11857:
11856:
11851:
11849:
11848:
11833:
11832:
11813:
11808:
11798:
11787:
11766:
11764:
11763:
11758:
11756:
11752:
11751:
11750:
11732:
11731:
11717:
11716:
11698:
11697:
11692:
11691:
11690:
11669:
11664:
11654:
11649:
11627:
11625:
11624:
11619:
11617:
11616:
11601:
11600:
11587:
11582:
11572:
11567:
11545:
11543:
11542:
11537:
11535:
11534:
11529:
11528:
11527:
11501:
11499:
11498:
11493:
11485:
11484:
11466:
11465:
11455:
11444:
11423:
11421:
11420:
11415:
11394:
11392:
11391:
11386:
11378:
11377:
11359:
11358:
11340:
11339:
11321:
11320:
11310:
11305:
11280:
11278:
11277:
11272:
11270:
11269:
11247:
11245:
11244:
11239:
11231:
11230:
11212:
11211:
11201:
11196:
11164:Induction proofs
11160:
11158:
11157:
11152:
11147:
11145:
11144:
11143:
11118:
11106:
11104:
11103:
11098:
11096:
11095:
11086:
11085:
11075:
11070:
11041:
11039:
11038:
11033:
11007:
11005:
11004:
10999:
10987:
10985:
10984:
10979:
10974:
10973:
10972:
10971:
10930:
10928:
10927:
10922:
10920:
10919:
10918:
10902:
10901:
10875:
10863:
10861:
10860:
10855:
10853:
10852:
10828:
10827:
10809:
10808:
10792:
10790:
10789:
10784:
10782:
10781:
10763:
10762:
10746:
10738:
10734:
10732:
10731:
10726:
10724:
10723:
10707:
10705:
10704:
10699:
10697:
10696:
10681:
10680:
10667:
10662:
10652:
10647:
10622:
10614:
10612:
10611:
10606:
10604:
10603:
10580:
10572:
10570:
10569:
10564:
10562:
10561:
10532:
10530:
10529:
10524:
10516:
10515:
10494:
10493:
10480:
10475:
10453:
10451:
10450:
10445:
10443:
10442:
10427:
10426:
10407:
10396:
10371:
10369:
10368:
10363:
10355:
10354:
10336:
10335:
10325:
10320:
10298:
10296:
10295:
10290:
10282:
10281:
10262:
10260:
10259:
10254:
10252:
10208:
10200:
10150:
10130:
10129:
10111:
10110:
10098:
10097:
10069:
10067:
10066:
10061:
9965:
9963:
9962:
9957:
9882:
9880:
9879:
9874:
9854:
9852:
9851:
9846:
9838:
9837:
9819:
9818:
9808:
9803:
9781:
9773:
9766:
9764:
9763:
9758:
9756:
9755:
9739:
9737:
9736:
9731:
9729:
9728:
9710:
9709:
9687:
9685:
9684:
9679:
9677:
9676:
9654:
9652:
9651:
9646:
9644:
9643:
9621:
9619:
9618:
9613:
9595:
9593:
9592:
9587:
9569:
9567:
9566:
9561:
9559:
9491:
9483:
9415:
9407:
9406:
9390:
9388:
9387:
9382:
9380:
9379:
9363:
9361:
9360:
9355:
9353:
9352:
9334:
9333:
9315:
9314:
9295:
9293:
9292:
9287:
9285:
9157:
9143:
9142:
9125:
9123:
9122:
9117:
9115:
9047:
9033:
9032:
9015:
9013:
9012:
9007:
9005:
8967:
8953:
8952:
8935:
8933:
8932:
8927:
8925:
8905:
8891:
8890:
8873:
8871:
8870:
8865:
8863:
8846:
8832:
8831:
8814:
8812:
8811:
8806:
8804:
8793:
8779:
8778:
8751:
8749:
8748:
8743:
8741:
8736:
8725:
8713:
8711:
8710:
8705:
8697:
8696:
8680:
8678:
8677:
8672:
8664:
8663:
8647:
8645:
8644:
8639:
8637:
8636:
8620:
8618:
8617:
8612:
8594:
8592:
8591:
8586:
8584:
8583:
8546:
8542:
8525:
8523:
8522:
8517:
8515:
8508:
8507:
8495:
8494:
8482:
8481:
8454:
8450:
8449:
8437:
8436:
8424:
8423:
8396:
8392:
8391:
8379:
8378:
8360:
8359:
8334:
8333:
8332:
8331:
8323:
8303:
8302:
8297:
8296:
8295:
8275:
8274:
8269:
8268:
8267:
8249:
8248:
8219:
8207:
8205:
8204:
8199:
8197:
8190:
8189:
8167:
8166:
8157:
8156:
8138:
8137:
8122:
8121:
8105:
8104:
8076:
8075:
8074:
8058:
8057:
8056:
8037:
8036:
8035:
8025:
8024:
8023:
8002:
7995:
7984:, the following
7983:
7974:
7962:Moreover, since
7959:
7957:
7956:
7951:
7946:
7945:
7940:
7939:
7938:
7924:
7923:
7908:
7907:
7889:
7888:
7852:
7841:
7835:
7833:
7832:
7827:
7822:
7821:
7814:
7813:
7796:
7795:
7782:
7781:
7770:
7769:
7743:
7742:
7737:
7736:
7691:
7670:
7662:
7658:
7628:
7618:
7616:
7615:
7610:
7605:
7603:
7601:
7600:
7598:
7596:
7595:
7593:
7591:
7575:
7573:
7563:
7549:
7547:
7537:
7523:
7521:
7511:
7488:
7469:
7461:
7459:
7458:
7453:
7448:
7446:
7444:
7439:
7432:
7430:
7429:
7428:
7404:
7403:
7388:
7383:
7382:
7364:
7362:
7361:
7356:
7354:
7347:
7346:
7345:
7332:
7325:
7324:
7302:
7301:
7276:
7270:
7266:
7264:
7263:
7256:
7255:
7220:
7219:
7202:
7201:
7182:
7181:
7147:
7143:
7142:
7141:
7135:
7134:
7125:
7124:
7114:
7107:
7106:
7094:
7093:
7086:
7085:
7050:
7049:
7023:
7019:
7018:
7017:
7011:
7010:
7001:
7000:
6990:
6983:
6982:
6970:
6969:
6951:
6947:
6946:
6945:
6939:
6938:
6929:
6928:
6918:
6911:
6910:
6894:
6893:
6892:
6886:
6885:
6876:
6875:
6859:
6839:
6833:
6831:
6830:
6825:
6820:
6819:
6800:
6799:
6761:
6760:
6752:
6751:
6694:
6692:
6691:
6686:
6684:
6677:
6676:
6664:
6663:
6644:
6643:
6627:
6626:
6581:
6569:
6567:
6566:
6561:
6556:
6555:
6549:
6545:
6543:
6541:
6531:
6529:
6528:
6523:
6509:
6502:
6500:
6498:
6493:
6486:
6484:
6474:
6468:
6467:
6461:
6457:
6455:
6453:
6443:
6441:
6440:
6435:
6421:
6414:
6412:
6410:
6405:
6398:
6396:
6386:
6381:
6380:
6360:
6354:
6352:
6351:
6346:
6344:
6337:
6336:
6335:
6326:
6325:
6324:
6307:
6300:
6299:
6293:
6289:
6287:
6285:
6275:
6273:
6272:
6267:
6253:
6246:
6244:
6242:
6237:
6230:
6228:
6218:
6211:
6210:
6209:
6196:
6189:
6188:
6182:
6178:
6176:
6174:
6164:
6162:
6161:
6156:
6142:
6135:
6133:
6131:
6126:
6119:
6117:
6107:
6099:
6095:
6094:
6086:
6083:
6082:
6061:
6055:
6051:
6046:
6045:
6037:
6034:
6033:
6024:
6018:
6014:
6006:
6002:
6001:
5993:
5990:
5989:
5980:
5974:
5970:
5965:
5964:
5956:
5953:
5952:
5943:
5937:
5933:
5924:
5923:
5918:
5917:
5909:
5894:
5888:
5886:
5885:
5880:
5875:
5874:
5866:
5863:
5857:
5853:
5848:
5847:
5839:
5836:
5830:
5826:
5821:
5820:
5819:
5806:
5796:
5795:
5790:
5789:
5781:
5768:
5766:
5765:
5760:
5755:
5754:
5753:
5744:
5743:
5742:
5725:
5715:
5714:
5706:
5699:
5698:
5697:
5684:
5674:
5673:
5665:
5652:
5650:
5649:
5644:
5642:
5641:
5632:
5631:
5626:
5618:
5617:
5611:
5602:
5596:
5595:
5567:
5565:
5564:
5559:
5557:
5556:
5547:
5546:
5541:
5533:
5532:
5526:
5517:
5501:
5488:
5486:
5485:
5480:
5478:
5477:
5472:
5471:
5463:
5459:
5458:
5453:
5444:
5443:
5438:
5437:
5429:
5416:
5414:
5413:
5408:
5403:
5402:
5397:
5396:
5388:
5384:
5376:
5375:
5364:
5363:
5355:
5344:
5342:
5341:
5336:
5334:
5333:
5332:
5326:
5325:
5316:
5315:
5299:
5292:
5291:
5253:
5252:
5251:
5245:
5244:
5229:
5228:
5212:
5181:
5179:
5178:
5173:
5167:
5166:
5157:
5156:
5135:
5134:
5119:
5118:
5094:
5093:
5088:
5087:
5086:
5062:
5060:
5059:
5054:
5049:
5041:
5040:
5019:
5018:
5006:
5001:
4999:
4998:
4983:
4982:
4961:
4960:
4948:
4947:
4929:
4927:
4926:
4921:
4916:
4911:
4910:
4908:
4907:
4883:
4882:
4877:
4876:
4875:
4861:
4856:
4851:
4849:
4848:
4838:
4833:
4832:
4810:
4808:
4807:
4802:
4800:
4799:
4780:and solved as a
4779:
4777:
4776:
4771:
4769:
4768:
4759:
4754:
4745:
4743:
4742:
4737:
4735:
4730:
4728:
4720:
4719:
4707:
4706:
4685:
4684:
4669:
4668:
4648:
4646:
4645:
4640:
4632:
4631:
4612:
4610:
4609:
4604:
4596:
4595:
4577:at least one of
4572:
4561:
4559:
4558:
4553:
4548:
4547:
4529:
4528:
4510:
4509:
4482:
4473:
4471:
4470:
4465:
4460:
4459:
4438:
4437:
4425:
4424:
4408:
4406:
4405:
4400:
4386:
4385:
4369:
4367:
4366:
4361:
4356:
4332:
4330:
4329:
4324:
4319:
4318:
4303:
4302:
4284:
4283:
4265:
4264:
4246:
4245:
4227:
4226:
4193:
4192:
4177:
4176:
4158:
4157:
4148:
4147:
4129:
4128:
4107:
4106:
4091:
4090:
4068:
4053:
4051:
4050:
4045:
4040:
4039:
4018:
4017:
4005:
4004:
3980:
3978:
3977:
3972:
3960:
3958:
3957:
3952:
3950:
3949:
3931:
3929:
3928:
3923:
3906:
3905:
3869:
3867:
3866:
3861:
3849:
3847:
3846:
3841:
3839:
3838:
3826:
3824:
3823:
3814:
3813:
3798:
3795:
3766:
3764:
3763:
3758:
3753:
3748:
3747:
3732:
3731:
3715:
3713:
3712:
3707:
3705:
3704:
3688:
3686:
3685:
3680:
3678:
3677:
3659:
3657:
3656:
3651:
3640:
3638:
3637:
3628:
3627:
3612:
3609:
3588:
3586:
3585:
3580:
3551:
3549:
3548:
3543:
3538:
3536:
3525:
3511:
3500:
3499:
3480:
3469:
3460:decimal digits.
3459:
3455:
3448:
3446:
3445:
3440:
3422:
3421:
3402:
3391:
3389:
3388:
3383:
3381:
3376:
3374:
3369:
3368:
3333:
3331:
3330:
3325:
3302:
3301:
3285:
3283:
3282:
3277:
3244:
3242:
3241:
3236:
3222:
3221:
3212:
3195:
3194:
3167:
3132:
3131:
3115:
3113:
3112:
3107:
3091:
3087:
3083:
3076:
3059:
3054:
3049:
3048:
3022:
3021:
3020:
2985:
2975:
2973:
2972:
2967:
2954:
2950:
2946:
2942:
2937:
2932:
2931:
2895:
2891:
2884:
2877:
2871:
2869:
2868:
2863:
2850:
2846:
2842:
2836:
2835:
2834:
2825:
2816:
2815:
2795:
2793:
2792:
2787:
2785:
2779:
2778:
2777:
2768:
2755:
2744:
2737:
2735:
2734:
2729:
2727:
2719:
2714:
2710:
2704:
2703:
2702:
2693:
2673:
2671:
2670:
2665:
2663:
2656:
2650:
2649:
2648:
2647:
2632:
2631:
2621:
2602:
2596:
2595:
2591:
2590:
2578:
2577:
2567:
2540:
2538:
2537:
2532:
2530:
2529:
2514:
2513:
2498:
2497:
2479:
2470:
2456:
2454:
2453:
2448:
2427:
2421:
2417:
2412:
2410:
2396:
2377:
2375:
2374:
2369:
2367:
2364:
2295:
2291:
2287:
2276:
2266:
2256:
2252:
2246:
2244:
2243:
2238:
2236:
2229:
2228:
2210:
2209:
2188:
2184:
2183:
2162:
2161:
2140:
2139:
2118:
2117:
2093:
2086:
2085:
2067:
2066:
2039:
2038:
2020:
2019:
1992:
1988:
1987:
1972:
1971:
1952:
1951:
1927:
1925:
1924:
1919:
1917:
1916:
1901:
1900:
1885:
1884:
1866:
1862:
1856:
1854:
1853:
1848:
1846:
1839:
1838:
1820:
1819:
1797:
1796:
1780:
1779:
1761:
1760:
1738:
1737:
1715:
1711:
1707:
1705:
1704:
1699:
1694:
1693:
1675:
1674:
1656:
1655:
1639:
1637:
1636:
1631:
1617:
1616:
1600:
1596:
1590:
1588:
1587:
1582:
1577:
1575:
1561:
1560:
1559:
1535:
1534:
1524:
1519:
1513:
1512:
1511:
1510:
1486:
1485:
1475:
1470:
1469:
1451:
1449:
1448:
1443:
1441:
1440:
1410:
1408:
1407:
1402:
1381:
1373:
1353:
1348:
1347:
1342:
1333:
1312:
1302:
1300:
1299:
1294:
1279:
1274:
1273:
1268:
1259:
1240:
1238:
1237:
1232:
1227:
1221:
1220:
1219:
1218:
1206:
1205:
1195:
1190:
1188:
1177:
1176:
1175:
1163:
1162:
1152:
1147:
1146:
1127:Daniel Bernoulli
1073:the end of the n
1052:
1048:
1044:
1037:
1030:
988:
970:
917:
905:
894:
882:
874:
863:
851:
846:Sanskrit prosody
835:
826:
817:
719:
709:
699:
689:
679:
669:
659:
649:
639:
629:
619:
609:
599:
589:
579:
569:
559:
549:
539:
529:
519:
518:
513:
501:
494:
492:
491:
486:
484:
483:
465:
464:
446:
445:
429:
427:
426:
421:
410:
409:
397:
396:
380:
378:
377:
372:
364:
363:
344:
337:
335:
334:
329:
327:
326:
308:
307:
289:
288:
272:
270:
269:
264:
253:
252:
233:
232:
177:
173:
169:
106:
62:
61:
58:
21:
28933:
28932:
28928:
28927:
28926:
28924:
28923:
28922:
28903:
28902:
28901:
28896:
28856:
28835:
28808:
28803:
28773:
28768:
28750:
28707:
28656:Kinds of series
28647:
28586:
28553:Explicit series
28544:
28518:
28480:
28466:Cauchy sequence
28454:
28441:
28395:Figurate number
28372:
28366:
28357:Powers of three
28301:
28292:
28262:
28257:
28248:
28181:Kepler triangle
28147:
28142:
28112:
28107:
28085:
28081:Strobogrammatic
28072:
28054:
28036:
28018:
28000:
27982:
27964:
27946:
27923:
27902:
27886:
27845:Divisor-related
27840:
27800:
27751:
27721:
27658:
27642:
27621:
27588:
27561:
27549:
27531:
27443:
27442:related numbers
27416:
27393:
27360:
27351:Perfect totient
27317:
27294:
27225:Highly abundant
27167:
27146:
27078:
27061:
27033:
27016:
27002:Stirling second
26908:
26885:
26846:
26828:
26785:
26734:
26671:
26632:Centered square
26600:
26583:
26545:
26530:
26497:
26482:
26434:
26433:defined numbers
26416:
26383:
26368:
26339:Double Mersenne
26325:
26306:
26228:
26214:
26212:natural numbers
26208:
26153:
26116:
26112:
26103:
26073:
26045:
26027:
25999:
25977:
25958:
25938:
25933:
25932:
25924:
25920:
25912:
25908:
25868:
25864:
25828:
25824:
25813:
25812:
25808:
25800:
25796:
25788:
25786:
25782:
25771:
25765:
25758:
25753:
25749:
25733:
25729:
25728:
25724:
25718:
25687:
25683:
25675:
25671:
25659:(3â4): 179â89,
25649:
25645:
25639:
25629:Springer-Verlag
25621:
25617:
25604:
25602:
25600:
25584:
25580:
25572:
25568:
25560:
25556:
25544:
25536:
25532:
25519:
25518:
25514:
25501:
25497:
25491:
25477:
25473:
25465:
25442:
25436:
25432:
25424:
25422:
25413:
25409:
25402:Multimedia Wiki
25398:
25397:
25393:
25384:
25383:
25379:
25368:
25364:
25339:
25335:
25329:
25315:Knuth, Donald E
25312:
25308:
25289:
25273:
25269:
25247:
25243:
25220:
25216:
25210:
25196:
25192:
25184:
25180:
25155:
25151:
25146:
25142:
25121:
25117:
25103:10.2307/2325076
25087:
25083:
25072:
25061:
25052:
25051:
25047:
25037:
25026:
25017:
25016:
25012:
25004:
25000:
24992:
24988:
24980:
24976:
24970:
24953:
24949:
24928:
24924:
24911:
24907:
24873:
24869:
24843:
24839:
24793:
24789:
24773:
24767:
24763:
24715:
24711:
24696:
24692:
24662:
24658:
24652:
24638:
24634:
24626:
24624:
24620:
24606:
24587:
24581:Diaconis, Persi
24578:
24574:
24553:
24549:
24541:
24537:
24476:
24464:
24460:
24458:
24455:
24454:
24451:
24447:
24439:
24437:
24424:
24420:
24406:
24402:
24396:
24378:
24374:
24350:
24346:
24329:
24321:
24317:
24305:
24301:
24279:10.2307/3618079
24273:(486): 521â25,
24263:
24259:
24238:
24229:
24223:
24207:
24203:
24197:
24183:
24179:
24171:
24167:
24160:
24149:
24145:
24136:
24134:
24132:nrich.maths.org
24126:
24125:
24121:
24112:
24110:
24096:
24090:
24086:
24077:
24076:
24072:
24066:
24052:
24048:
24027:
24023:
24002:
23998:
23990:
23986:
23978:
23974:
23968:
23946:
23942:
23932:
23916:
23912:
23901:
23887:Gardner, Martin
23884:
23880:
23863:
23859:
23850:
23848:
23836:
23832:
23826:
23812:
23808:
23799:
23797:
23786:
23785:
23781:
23773:
23769:
23760:
23758:
23748:
23742:
23738:
23730:
23726:
23715:
23711:
23681:
23677:
23659:
23655:
23647:
23640:
23629:
23610:
23606:
23598:
23594:
23586:
23582:
23574:
23570:
23562:
23558:
23546:
23541:
23522:
23515:
23491:
23484:
23478:
23462:
23453:
23432:
23425:
23420:
23415:
23414:
23405:
23401:
23396:
23391:
23355:
23318:resistor ladder
23310:
23274:
23270:
23263:
23259:
23247:
23213:
23209:
23207:
23204:
23203:
23201:
23176:
23172:
23170:
23167:
23166:
23164:
23139:
23135:
23133:
23130:
23129:
23127:
23102:
23098:
23096:
23093:
23092:
23090:
23067:
23063:
23061:
23058:
23057:
23034:
23025:
23022:
23013:
23010:
23002:
22971:
22967:
22966:for some index
22952:
22948:
22929:
22918:Fermat's spiral
22913:
22909:
22889:
22863:
22859:
22851:
22849:
22841:
22838:
22837:
22827:
22806:
22774:Alexander Braun
22719:
22713:
22707:
22646:data structure.
22604:
22577:
22574:Balance factors
22566:
22505:
22489:
22485:
22484:
22483:
22474:
22470:
22458:
22454:
22442:
22438:
22423:
22419:
22407:
22403:
22397:
22393:
22388:
22385:
22384:
22339:, which led to
22326:
22322:
22318:
22311:
22308:
22299:
22295:
22291:
22287:
22280:
22277:
22268:
22260:
22256:
22252:
22248:
22242:
22239:
22230:
22226:
22222:
22212:
22204:
22189:
22185:
22177:
22173:
22170:
22161:
22153:
22149:
22143:
22140:
22131:
22127:
22123:
22089:
22084:
22079:
22070:
22065:
22060:
22055:
22050:
22045:
22034:
22027:
22021:
21999:
21996:
21995:
21988:
21979:
21971:
21963:
21945:
21941:
21922:
21909:
21908:
21907:
21897:
21884:
21883:
21882:
21872:
21859:
21858:
21857:
21854:
21851:
21850:
21839:
21834:
21829:
21820:
21815:
21810:
21805:
21796:
21791:
21764:
21760:
21758:
21755:
21754:
21737:
21733:
21727:
21723:
21717:
21706:
21687:
21683:
21659:
21655:
21640:
21636:
21618:
21614:
21590:
21586:
21568:
21564:
21551:
21546:
21544:
21541:
21540:
21519:
21498:
21492:
21491:
21490:
21468:
21466:
21462:
21461:
21450:
21437:
21433:
21431:
21428:
21427:
21404:
21399:
21359:
21340:
21337:
21327:
21316:
21311:
21297:
21287:
21276:
21271:
21267:
21261:
21257:
21251:
21211:
21205:
21203:Generalizations
21197:cycle detection
21192:
21185:modular integer
21168:
21161:
21153:
21150:
21144:
21129:
21118:
21080:
21071:
21067:
21049:
21045:
21027:
21023:
20998:
20989:
20985:
20967:
20963:
20945:
20941:
20923:
20919:
20901:
20897:
20895:
20892:
20891:
20885:
20877:
20874:
20866:
20862:
20833:
20815:
20808:
20804:
20803:
20802:
20794: and
20792:
20774:
20756:
20749:
20745:
20744:
20743:
20738:
20735:
20734:
20712:
20708:
20705: and
20703:
20691:
20687:
20685:
20682:
20681:
20648:
20647:
20635:
20629:
20628:
20624:
20620:
20608:
20581:
20580:
20568:
20562:
20561:
20557:
20553:
20541:
20522:
20521:
20509:
20503:
20502:
20500:
20497:
20496:
20489:
20488:, in this case
20482:
20451:
20433:
20426:
20422:
20421:
20420:
20412: and
20410:
20392:
20371:
20364:
20360:
20359:
20358:
20353:
20350:
20349:
20327:
20323:
20320: and
20318:
20306:
20302:
20300:
20297:
20296:
20263:
20262:
20250:
20244:
20243:
20239:
20235:
20223:
20193:
20192:
20180:
20174:
20173:
20169:
20165:
20153:
20134:
20133:
20121:
20115:
20114:
20112:
20109:
20108:
20101:
20100:, in this case
20094:
20063:
20045:
20038:
20034:
20033:
20032:
20024: and
20022:
20004:
19986:
19979:
19975:
19974:
19973:
19968:
19965:
19964:
19942:
19938:
19935: and
19933:
19921:
19917:
19915:
19912:
19911:
19878:
19877:
19865:
19859:
19858:
19854:
19850:
19838:
19811:
19810:
19798:
19792:
19791:
19787:
19783:
19771:
19752:
19751:
19739:
19733:
19732:
19730:
19727:
19726:
19719:
19718:, in this case
19712:
19681:
19660:
19653:
19649:
19648:
19647:
19639: and
19637:
19619:
19598:
19591:
19587:
19586:
19585:
19580:
19577:
19576:
19554:
19550:
19547: and
19545:
19533:
19529:
19527:
19524:
19523:
19487:
19486:
19474:
19468:
19467:
19463:
19459:
19447:
19417:
19416:
19404:
19398:
19397:
19393:
19389:
19377:
19358:
19357:
19345:
19339:
19338:
19336:
19333:
19332:
19325:
19324:, in this case
19318:
19297:
19296:
19277:
19263:
19261:
19245:
19228:
19227:
19215:
19209:
19208:
19204:
19200:
19188:
19185:
19184:
19168:
19154:
19152:
19136:
19119:
19118:
19106:
19100:
19099:
19095:
19091:
19079:
19072:
19071:
19062:
19044:
19042:
19038:
19037:
19036:
19031:
19028:
19027:
19020:
18987:
18983:
18974:
18952:
18948:
18941:
18937:
18935:
18932:
18931:
18925:
18909:
18908:
18898:
18892:
18888:
18886:
18873:
18867:
18863:
18861:
18845:
18839:
18838:
18826:
18820:
18819:
18816:
18815:
18802:
18796:
18792:
18790:
18777:
18771:
18767:
18765:
18749:
18743:
18742:
18730:
18724:
18723:
18720:
18719:
18706:
18700:
18696:
18694:
18681:
18675:
18674:
18662:
18656:
18655:
18652:
18651:
18638:
18632:
18628:
18626:
18613:
18607:
18603:
18601:
18585:
18579:
18578:
18566:
18560:
18559:
18556:
18555:
18542:
18536:
18532:
18530:
18517:
18511:
18507:
18505:
18489:
18483:
18482:
18470:
18464:
18463:
18459:
18457:
18454:
18453:
18421:
18399:
18395:
18388:
18384:
18378:
18361:
18347:
18343:
18334:
18330:
18328:
18325:
18324:
18320:
18301:
18300:
18281:
18264:
18262:
18253:
18252:
18236:
18219:
18217:
18211:
18210:
18196:
18194:
18184:
18183:
18166:
18162:
18160:
18157:
18156:
18139:
18138:
18126:
18120:
18119:
18117:
18114:
18113:
18111:Legendre symbol
18106:
18100:
18094:
18091:
18085:
18078:
18072:
18069:
18063:
18060:
18053:
18047:
18044:
17998:
17992:
17977:
17971:
17968:
17960:
17957:
17949:
17932:Fibonacci prime
17928:
17926:Fibonacci prime
17922:
17912:, which can be
17903:
17873:
17867:
17860:
17859:
17854:
17848:
17847:
17842:
17832:
17831:
17830:
17820:
17819:
17807:
17803:
17801:
17795:
17791:
17788:
17787:
17781:
17777:
17775:
17763:
17759:
17752:
17751:
17749:
17746:
17745:
17735:
17727:
17719:
17711:
17703:
17699:
17695:
17661:
17657:
17648:
17644:
17636:
17633:
17632:
17625:
17591:
17587:
17578:
17574:
17566:
17563:
17562:
17559:Legendre symbol
17535:
17534:
17519:
17515:
17504:
17488:
17473:
17472:
17457:
17453:
17442:
17426:
17411:
17410:
17401:
17397:
17386:
17370:
17369:
17367:
17364:
17363:
17358:
17350:
17343:
17340:
17331:
17327:
17323:
17320:
17311:
17307:
17303:
17296:
17292:
17282:
17247:
17243:
17228:
17224:
17200:
17196:
17187:
17183:
17159:
17155:
17146:
17142:
17134:
17131:
17130:
17104:
17100:
17098:
17095:
17094:
17071:
17067:
17065:
17062:
17061:
17046:
17001:
16997:
16979:
16975:
16966:
16962:
16953:
16949:
16941:
16938:
16937:
16930:
16922:
16899:
16895:
16893:
16890:
16889:
16886:
16881:
16853:
16849:
16848:
16844:
16830:
16826:
16825:
16821:
16819:
16800:
16796:
16795:
16791:
16786:
16780:
16769:
16763:
16760:
16759:
16755:
16728:
16721:
16719:
16706:
16702:
16701:
16697:
16692:
16686:
16675:
16669:
16666:
16665:
16662:Millin's series
16618:
16614:
16609:
16603:
16592:
16568:
16564:
16559:
16553:
16542:
16527:
16523:
16518:
16512:
16501:
16495:
16492:
16491:
16452:
16448:
16441:
16432:
16428:
16426:
16411:
16407:
16400:
16391:
16387:
16385:
16384:
16380:
16373:
16359:
16355:
16350:
16347:
16344:
16343:
16319:
16315:
16308:
16302:
16298:
16296:
16290:
16279:
16257:
16254:
16253:
16225:
16221:
16203:
16199:
16192:
16188:
16181:
16167:
16163:
16158:
16152:
16141:
16135:
16132:
16131:
16096:
16095:
16093:
16081:
16074:
16070:
16069:
16068:
16062:
16051:
16046:
16033:
16029:
16019:
16017:
16011:
16000:
15994:
15991:
15990:
15981:and there is a
15954:
15933:
15929:
15922:
15917:
15911:
15900:
15894:
15891:
15890:
15857:
15839:
15832:
15830:
15823:
15819:
15818:
15811:
15807:
15798:
15780:
15773:
15771:
15764:
15760:
15759:
15752:
15748:
15747:
15743:
15732:
15721:
15714:
15710:
15709:
15708:
15703:
15697:
15686:
15680:
15677:
15676:
15654:
15636:
15629:
15627:
15620:
15616:
15615:
15608:
15604:
15591:
15571:
15567:
15562:
15556:
15545:
15539:
15536:
15535:
15532:theta functions
15524:
15522:Reciprocal sums
15487:
15474:
15457:
15454:
15453:
15428:
15425:
15424:
15408:
15405:
15404:
15373:
15369:
15365:
15344:
15341:
15340:
15310:
15307:
15306:
15278:
15271:
15267:
15253:
15250:
15249:
15219:
15212:
15208:
15196:
15188:
15185:
15184:
15161:
15154:
15150:
15138:
15130:
15127:
15126:
15094:
15089:
15070:
15065:
15064:
15060:
15048:
15031:
15028:
15027:
14998:
14995:
14994:
14977:
14973:
14971:
14968:
14967:
14951:
14950:
14935:
14934:
14928:
14924:
14909:
14905:
14890:
14886:
14877:
14873:
14864:
14853:
14840:
14836:
14824:
14820:
14808:
14804:
14792:
14791:
14785:
14781:
14769:
14765:
14759:
14748:
14735:
14731:
14719:
14715:
14709:
14698:
14685:
14681:
14675:
14671:
14665:
14654:
14641:
14640:
14628:
14624:
14618:
14614:
14608:
14597:
14578:
14574:
14568:
14564:
14558:
14547:
14534:
14530:
14524:
14520:
14514:
14503:
14492:
14471:
14467:
14448:
14446:
14443:
14442:
14422:
14418:
14401:
14398:
14397:
14372:
14368:
14355:
14350:
14333:
14330:
14329:
14305:
14294:
14286:
14284:
14281:
14280:
14264:
14261:
14260:
14229:
14225:
14213:
14209:
14197:
14193:
14181:
14177:
14156:
14152:
14146:
14142:
14136:
14125:
14104:
14101:
14100:
14088:
14072:
14044:
14031:
14027:
14026:
14025:
14019:
14012:
14008:
14007:
14006:
13994:
13990:
13983:
13970:
13969:
13968:
13962:
13951:
13929:
13925:
13923:
13920:
13919:
13889:
13876:
13872:
13871:
13870:
13864:
13857:
13853:
13852:
13851:
13839:
13835:
13828:
13815:
13814:
13813:
13807:
13796:
13774:
13770:
13768:
13765:
13764:
13725:
13712:
13708:
13707:
13706:
13694:
13687:
13683:
13682:
13681:
13680:
13676:
13670:
13663:
13659:
13658:
13657:
13640:
13633:
13629:
13628:
13627:
13615:
13602:
13598:
13597:
13596:
13595:
13591:
13579:
13575:
13569:
13565:
13550:
13546:
13544:
13541:
13540:
13524:
13517:
13513:
13512:
13511:
13502:
13498:
13492:
13479:
13475:
13474:
13473:
13461:
13448:
13444:
13443:
13442:
13424:
13420:
13418:
13415:
13414:
13398:
13391:
13387:
13386:
13385:
13376:
13369:
13365:
13364:
13363:
13351:
13347:
13335:
13322:
13318:
13317:
13316:
13298:
13294:
13292:
13289:
13288:
13269:
13265:
13259:
13255:
13234:
13227:
13223:
13222:
13221:
13203:
13199:
13187:
13183:
13177:
13173:
13161:
13154:
13150:
13149:
13148:
13133:
13129:
13127:
13124:
13123:
13119:
13111:
13108:
13100:
13083:
13079:
13073:
13069:
13049:
13045:
13030:
13026:
13025:
13021:
13015:
13011:
13002:
12989:
12985:
12984:
12983:
12974:
12961:
12957:
12956:
12955:
12943:
12939:
12937:
12934:
12933:
12911:
12907:
12901:
12897:
12879:
12875:
12863:
12859:
12844:
12840:
12834:
12830:
12828:
12825:
12824:
12822:
12801:
12794:
12790:
12789:
12788:
12776:
12772:
12748:
12744:
12732:
12728:
12719:
12712:
12708:
12707:
12706:
12704:
12701:
12700:
12677:
12673:
12649:
12645:
12633:
12629:
12620:
12613:
12609:
12608:
12607:
12605:
12602:
12601:
12598:
12592:
12584:
12561:
12547:
12544:
12543:
12515:
12512:
12511:
12497:
12496:
12475:
12471:
12461:
12452:
12451:
12421:
12417:
12402:
12398:
12389:
12388:
12378:
12359:
12355:
12334:
12330:
12326:
12324:
12315:
12314:
12304:
12285:
12281:
12280:
12278:
12265:
12246:
12242:
12241:
12239:
12230:
12229:
12216:
12197:
12193:
12192:
12190:
12174:
12155:
12151:
12150:
12148:
12139:
12138:
12127:
12123:
12108:
12104:
12091:
12087:
12078:
12074:
12059:
12055:
12047:
12037:
12035:
12032:
12031:
12013:
12011:
12008:
12007:
11978:
11974:
11965:
11961:
11955:
11944:
11938:
11935:
11934:
11911:
11907:
11898:
11894:
11885:
11881:
11874:
11872:
11869:
11868:
11865:
11838:
11834:
11822:
11818:
11809:
11804:
11788:
11777:
11771:
11768:
11767:
11740:
11736:
11727:
11723:
11722:
11718:
11706:
11702:
11693:
11680:
11676:
11675:
11674:
11665:
11660:
11650:
11639:
11633:
11630:
11629:
11606:
11602:
11596:
11592:
11583:
11578:
11568:
11557:
11551:
11548:
11547:
11530:
11517:
11513:
11512:
11511:
11509:
11506:
11505:
11504:Similarly, add
11474:
11470:
11461:
11457:
11445:
11434:
11428:
11425:
11424:
11403:
11400:
11399:
11367:
11363:
11348:
11344:
11329:
11325:
11316:
11312:
11306:
11295:
11289:
11286:
11285:
11259:
11255:
11253:
11250:
11249:
11220:
11216:
11207:
11203:
11197:
11186:
11180:
11177:
11176:
11166:
11139:
11135:
11122:
11117:
11115:
11112:
11111:
11091:
11087:
11081:
11077:
11071:
11060:
11054:
11051:
11050:
11021:
11018:
11017:
10993:
10990:
10989:
10967:
10963:
10956:
10955:
10944:
10941:
10940:
10933:symbolic method
10914:
10907:
10903:
10897:
10893:
10888:
10885:
10884:
10881:
10879:Symbolic method
10848:
10844:
10817:
10813:
10804:
10800:
10798:
10795:
10794:
10771:
10767:
10758:
10754:
10752:
10749:
10748:
10740:
10736:
10719:
10715:
10713:
10710:
10709:
10686:
10682:
10676:
10672:
10663:
10658:
10648:
10637:
10631:
10628:
10627:
10616:
10596:
10592:
10590:
10587:
10586:
10574:
10548:
10544:
10542:
10539:
10538:
10502:
10498:
10486:
10482:
10476:
10465:
10459:
10456:
10455:
10435:
10431:
10413:
10409:
10397:
10386:
10380:
10377:
10376:
10344:
10340:
10331:
10327:
10321:
10310:
10304:
10301:
10300:
10277:
10273:
10271:
10268:
10267:
10248:
10204:
10196:
10146:
10119:
10115:
10106:
10102:
10087:
10083:
10081:
10078:
10077:
9971:
9968:
9967:
9888:
9885:
9884:
9862:
9859:
9858:
9827:
9823:
9814:
9810:
9804:
9793:
9787:
9784:
9783:
9775:
9771:
9751:
9747:
9745:
9742:
9741:
9718:
9714:
9699:
9695:
9693:
9690:
9689:
9666:
9662:
9660:
9657:
9656:
9633:
9629:
9627:
9624:
9623:
9601:
9598:
9597:
9575:
9572:
9571:
9555:
9487:
9479:
9411:
9402:
9398:
9396:
9393:
9392:
9375:
9371:
9369:
9366:
9365:
9342:
9338:
9323:
9319:
9310:
9306:
9304:
9301:
9300:
9281:
9153:
9138:
9134:
9132:
9129:
9128:
9111:
9043:
9028:
9024:
9022:
9019:
9018:
9001:
8963:
8948:
8944:
8942:
8939:
8938:
8921:
8901:
8886:
8882:
8880:
8877:
8876:
8859:
8842:
8827:
8823:
8821:
8818:
8817:
8800:
8789:
8774:
8770:
8768:
8765:
8764:
8737:
8726:
8721:
8719:
8716:
8715:
8692:
8688:
8686:
8683:
8682:
8659:
8655:
8653:
8650:
8649:
8632:
8628:
8626:
8623:
8622:
8600:
8597:
8596:
8579:
8575:
8573:
8570:
8569:
8562:
8557:
8544:
8533:
8513:
8512:
8503:
8499:
8490:
8486:
8471:
8467:
8452:
8451:
8445:
8441:
8432:
8428:
8413:
8409:
8394:
8393:
8387:
8383:
8368:
8364:
8349:
8345:
8335:
8322:
8320:
8319:
8312:
8308:
8305:
8304:
8298:
8285:
8281:
8280:
8279:
8270:
8263:
8259:
8258:
8257:
8250:
8235:
8231:
8227:
8225:
8222:
8221:
8211:
8195:
8194:
8179:
8175:
8168:
8162:
8158:
8146:
8142:
8127:
8123:
8117:
8113:
8110:
8109:
8088:
8084:
8077:
8064:
8060:
8059:
8046:
8042:
8041:
8031:
8027:
8026:
8019:
8015:
8014:
8010:
8008:
8005:
8004:
7997:
7993:
7979:
7963:
7941:
7934:
7930:
7929:
7928:
7913:
7909:
7897:
7893:
7884:
7880:
7869:
7866:
7865:
7843:
7839:
7816:
7815:
7803:
7799:
7797:
7791:
7787:
7784:
7783:
7777:
7773:
7771:
7759:
7755:
7748:
7747:
7738:
7731:
7730:
7725:
7719:
7718:
7713:
7703:
7702:
7701:
7699:
7696:
7695:
7690:
7681:
7672:
7664:
7660:
7657:
7648:
7638:
7630:
7626:
7581:
7576:
7569:
7564:
7562:
7555:
7550:
7543:
7538:
7536:
7529:
7524:
7517:
7512:
7510:
7496:
7493:
7492:
7486:
7465:
7438:
7433:
7421:
7417:
7399:
7395:
7394:
7389:
7387:
7378:
7374:
7372:
7369:
7368:
7352:
7351:
7341:
7328:
7327:
7326:
7319:
7318:
7313:
7304:
7303:
7294:
7290:
7288:
7278:
7277:
7265:
7258:
7257:
7248:
7244:
7233:
7227:
7226:
7221:
7215:
7211:
7204:
7203:
7196:
7195:
7190:
7184:
7183:
7174:
7170:
7165:
7155:
7154:
7145:
7144:
7137:
7130:
7126:
7120:
7116:
7110:
7109:
7108:
7099:
7095:
7088:
7087:
7078:
7074:
7063:
7057:
7056:
7051:
7045:
7041:
7034:
7033:
7021:
7020:
7013:
7006:
7002:
6996:
6992:
6986:
6985:
6984:
6975:
6971:
6965:
6961:
6949:
6948:
6941:
6934:
6930:
6924:
6920:
6914:
6913:
6912:
6906:
6902:
6895:
6888:
6881:
6877:
6865:
6861:
6855:
6854:
6853:
6849:
6847:
6844:
6843:
6837:
6814:
6813:
6808:
6802:
6801:
6792:
6788:
6783:
6773:
6772:
6755:
6754:
6744:
6740:
6735:
6729:
6728:
6723:
6713:
6712:
6704:
6701:
6700:
6682:
6681:
6669:
6665:
6659:
6655:
6645:
6639:
6635:
6632:
6631:
6619:
6615:
6602:
6595:
6593:
6590:
6589:
6577:
6574:diagonalization
6550:
6537:
6532:
6522:
6515:
6510:
6508:
6504:
6503:
6492:
6487:
6480:
6475:
6473:
6462:
6449:
6444:
6434:
6427:
6422:
6420:
6416:
6415:
6404:
6399:
6392:
6387:
6385:
6376:
6372:
6370:
6367:
6366:
6358:
6357:From this, the
6342:
6341:
6331:
6317:
6313:
6309:
6303:
6302:
6301:
6294:
6281:
6276:
6266:
6259:
6254:
6252:
6248:
6247:
6236:
6231:
6224:
6219:
6217:
6205:
6192:
6191:
6190:
6183:
6170:
6165:
6155:
6148:
6143:
6141:
6137:
6136:
6125:
6120:
6113:
6108:
6106:
6097:
6096:
6085:
6084:
6075:
6071:
6050:
6036:
6035:
6029:
6025:
6013:
6004:
6003:
5992:
5991:
5985:
5981:
5969:
5955:
5954:
5948:
5944:
5932:
5925:
5919:
5908:
5907:
5906:
5902:
5900:
5897:
5896:
5892:
5865:
5864:
5852:
5838:
5837:
5825:
5815:
5802:
5801:
5800:
5791:
5780:
5779:
5778:
5776:
5773:
5772:
5749:
5735:
5731:
5727:
5721:
5720:
5719:
5705:
5704:
5693:
5680:
5679:
5678:
5664:
5663:
5661:
5658:
5657:
5637:
5636:
5625:
5613:
5612:
5600:
5588:
5584:
5573:
5570:
5569:
5552:
5551:
5540:
5528:
5527:
5515:
5507:
5504:
5503:
5497:
5473:
5462:
5461:
5460:
5454:
5449:
5448:
5439:
5428:
5427:
5426:
5424:
5421:
5420:
5398:
5387:
5386:
5385:
5380:
5365:
5354:
5353:
5352:
5350:
5347:
5346:
5328:
5321:
5317:
5305:
5301:
5295:
5294:
5293:
5286:
5285:
5280:
5274:
5273:
5268:
5258:
5257:
5247:
5234:
5230:
5218:
5214:
5208:
5207:
5206:
5204:
5201:
5200:
5191:
5162:
5158:
5146:
5142:
5130:
5126:
5114:
5110:
5089:
5082:
5078:
5077:
5076:
5071:
5068:
5067:
5045:
5030:
5026:
5014:
5010:
5000:
4994:
4990:
4972:
4968:
4956:
4952:
4943:
4939:
4937:
4934:
4933:
4903:
4899:
4878:
4871:
4867:
4866:
4865:
4860:
4850:
4844:
4840:
4839:
4837:
4828:
4824:
4822:
4819:
4818:
4795:
4791:
4789:
4786:
4785:
4764:
4760:
4753:
4751:
4748:
4747:
4729:
4724:
4712:
4708:
4702:
4698:
4680:
4676:
4664:
4660:
4658:
4655:
4654:
4627:
4623:
4618:
4615:
4614:
4591:
4587:
4582:
4579:
4578:
4570:
4567:
4537:
4533:
4518:
4514:
4505:
4501:
4499:
4496:
4495:
4488:
4477:
4449:
4445:
4433:
4429:
4420:
4416:
4414:
4411:
4410:
4381:
4377:
4375:
4372:
4371:
4352:
4338:
4335:
4334:
4314:
4310:
4292:
4288:
4279:
4275:
4254:
4250:
4241:
4237:
4216:
4212:
4188:
4184:
4166:
4162:
4153:
4149:
4143:
4139:
4118:
4114:
4102:
4098:
4080:
4076:
4074:
4071:
4070:
4063:
4029:
4025:
4013:
4009:
4000:
3996:
3994:
3991:
3990:
3966:
3963:
3962:
3945:
3941:
3939:
3936:
3935:
3901:
3897:
3895:
3892:
3891:
3888:
3855:
3852:
3851:
3834:
3830:
3819:
3815:
3803:
3799:
3797:
3785:
3779:
3776:
3775:
3769:Binet's formula
3749:
3743:
3739:
3727:
3723:
3721:
3718:
3717:
3700:
3696:
3694:
3691:
3690:
3673:
3669:
3667:
3664:
3663:
3633:
3629:
3617:
3613:
3611:
3599:
3593:
3590:
3589:
3571:
3568:
3567:
3560:Johannes Kepler
3557:
3526:
3512:
3510:
3495:
3491:
3486:
3483:
3482:
3479:
3471:
3467:
3457:
3450:
3417:
3413:
3408:
3405:
3404:
3401:
3393:
3375:
3370:
3364:
3360:
3358:
3355:
3354:
3347:
3340:
3297:
3293:
3291:
3288:
3287:
3250:
3247:
3246:
3217:
3213:
3208:
3190:
3186:
3163:
3127:
3123:
3121:
3118:
3117:
3072:
3053:
3044:
3040:
3039:
3035:
2998:
2997:
2993:
2991:
2988:
2987:
2983:
2936:
2927:
2923:
2922:
2918:
2901:
2898:
2897:
2893:
2886:
2879:
2875:
2830:
2826:
2824:
2820:
2811:
2807:
2805:
2802:
2801:
2773:
2769:
2767:
2765:
2762:
2761:
2756:is the closest
2754:
2746:
2739:
2718:
2698:
2694:
2692:
2688:
2686:
2683:
2682:
2679:
2661:
2660:
2643:
2639:
2627:
2623:
2622:
2620:
2613:
2607:
2606:
2586:
2582:
2573:
2569:
2568:
2566:
2559:
2552:
2550:
2547:
2546:
2525:
2521:
2509:
2505:
2493:
2489:
2487:
2484:
2483:
2478:
2472:
2469:
2463:
2416:
2400:
2395:
2387:
2384:
2383:
2362:
2361:
2351:
2333:
2332:
2322:
2309:
2305:
2303:
2300:
2299:
2293:
2289:
2286:
2278:
2274:
2268:
2264:
2258:
2254:
2250:
2234:
2233:
2218:
2214:
2199:
2195:
2186:
2185:
2173:
2169:
2151:
2147:
2129:
2125:
2107:
2103:
2091:
2090:
2075:
2071:
2056:
2052:
2028:
2024:
2009:
2005:
1990:
1989:
1983:
1979:
1967:
1963:
1953:
1947:
1943:
1939:
1937:
1934:
1933:
1912:
1908:
1896:
1892:
1880:
1876:
1874:
1871:
1870:
1864:
1860:
1844:
1843:
1828:
1824:
1809:
1805:
1798:
1792:
1788:
1785:
1784:
1769:
1765:
1750:
1746:
1739:
1733:
1729:
1725:
1723:
1720:
1719:
1713:
1709:
1683:
1679:
1664:
1660:
1651:
1647:
1645:
1642:
1641:
1612:
1608:
1606:
1603:
1602:
1598:
1594:
1562:
1552:
1548:
1530:
1526:
1525:
1523:
1503:
1499:
1481:
1477:
1476:
1474:
1465:
1461:
1459:
1456:
1455:
1433:
1429:
1418:
1415:
1414:
1372:
1341:
1334:
1332:
1324:
1321:
1320:
1310:
1267:
1260:
1258:
1250:
1247:
1246:
1214:
1210:
1201:
1197:
1196:
1194:
1178:
1171:
1167:
1158:
1154:
1153:
1151:
1142:
1138:
1136:
1133:
1132:
1115:Binet's formula
1099:
1094:
1088:
1080:
1050:
1046:
1039:
1032:
1028:
976:
953:
916:
907:
904:
896:
893:
884:
880:
862:
853:
849:
834:
828:
825:
819:
816:
810:
803:
797:
792:
718:
712:
708:
702:
698:
692:
688:
682:
678:
672:
668:
662:
658:
652:
648:
642:
638:
632:
628:
622:
618:
612:
608:
602:
598:
592:
588:
582:
578:
572:
568:
562:
558:
552:
548:
542:
538:
532:
528:
522:
511:
506:
496:
473:
469:
454:
450:
441:
437:
435:
432:
431:
405:
401:
392:
388:
386:
383:
382:
359:
355:
353:
350:
349:
339:
316:
312:
297:
293:
284:
280:
278:
275:
274:
248:
244:
228:
224:
222:
219:
218:
196:
188:Lucas sequences
175:
171:
167:
164:Binet's formula
133:Fibonacci cubes
59:
56:
51:
50:
35:
28:
23:
22:
15:
12:
11:
5:
28931:
28921:
28920:
28915:
28898:
28897:
28895:
28894:
28893:
28892:
28880:
28875:
28870:
28864:
28862:
28858:
28857:
28855:
28854:
28849:
28843:
28841:
28837:
28836:
28834:
28833:
28825:
28816:
28814:
28810:
28809:
28802:
28801:
28794:
28787:
28779:
28770:
28769:
28767:
28766:
28755:
28752:
28751:
28749:
28748:
28743:
28738:
28733:
28728:
28723:
28717:
28715:
28709:
28708:
28706:
28705:
28700:
28698:Fourier series
28695:
28690:
28685:
28683:Puiseux series
28680:
28678:Laurent series
28675:
28670:
28665:
28659:
28657:
28653:
28652:
28649:
28648:
28646:
28645:
28640:
28635:
28630:
28625:
28620:
28615:
28610:
28605:
28600:
28594:
28592:
28588:
28587:
28585:
28584:
28579:
28574:
28569:
28563:
28561:
28554:
28550:
28549:
28546:
28545:
28543:
28542:
28537:
28532:
28526:
28524:
28520:
28519:
28517:
28516:
28511:
28506:
28501:
28495:
28493:
28486:
28482:
28481:
28479:
28478:
28473:
28468:
28462:
28460:
28456:
28455:
28448:
28446:
28443:
28442:
28440:
28439:
28438:
28437:
28427:
28422:
28417:
28412:
28407:
28402:
28397:
28392:
28387:
28381:
28379:
28368:
28367:
28365:
28364:
28359:
28354:
28349:
28344:
28339:
28334:
28329:
28324:
28318:
28316:
28309:
28303:
28302:
28291:
28290:
28283:
28276:
28268:
28259:
28258:
28251:
28249:
28247:
28246:
28243:
28240:
28237:
28234:
28233:
28232:
28227:
28216:
28215:
28214:
28213:
28208:
28203:
28198:
28196:Section search
28193:
28188:
28183:
28178:
28173:
28168:
28158:
28152:
28149:
28148:
28145:Metallic means
28141:
28140:
28133:
28126:
28118:
28109:
28108:
28106:
28105:
28094:
28091:
28090:
28087:
28086:
28084:
28083:
28077:
28074:
28073:
28060:
28059:
28056:
28055:
28053:
28052:
28047:
28041:
28038:
28037:
28024:
28023:
28020:
28019:
28017:
28016:
28014:Sorting number
28011:
28009:Pancake number
28005:
28002:
28001:
27988:
27987:
27984:
27983:
27981:
27980:
27975:
27969:
27966:
27965:
27952:
27951:
27948:
27947:
27945:
27944:
27939:
27934:
27928:
27925:
27924:
27921:Binary numbers
27912:
27911:
27908:
27907:
27904:
27903:
27901:
27900:
27894:
27892:
27888:
27887:
27885:
27884:
27879:
27874:
27869:
27864:
27859:
27854:
27848:
27846:
27842:
27841:
27839:
27838:
27833:
27828:
27823:
27818:
27812:
27810:
27802:
27801:
27799:
27798:
27793:
27788:
27783:
27778:
27773:
27768:
27762:
27760:
27753:
27752:
27750:
27749:
27748:
27747:
27736:
27734:
27731:P-adic numbers
27727:
27726:
27723:
27722:
27720:
27719:
27718:
27717:
27707:
27702:
27697:
27692:
27687:
27682:
27677:
27672:
27666:
27664:
27660:
27659:
27657:
27656:
27650:
27648:
27647:Coding-related
27644:
27643:
27641:
27640:
27635:
27629:
27627:
27623:
27622:
27620:
27619:
27614:
27609:
27604:
27598:
27596:
27587:
27586:
27585:
27584:
27582:Multiplicative
27579:
27568:
27566:
27551:
27550:
27546:Numeral system
27537:
27536:
27533:
27532:
27530:
27529:
27524:
27519:
27514:
27509:
27504:
27499:
27494:
27489:
27484:
27479:
27474:
27469:
27464:
27459:
27454:
27448:
27445:
27444:
27426:
27425:
27422:
27421:
27418:
27417:
27415:
27414:
27409:
27403:
27401:
27395:
27394:
27392:
27391:
27386:
27381:
27376:
27370:
27368:
27362:
27361:
27359:
27358:
27353:
27348:
27343:
27338:
27336:Highly totient
27333:
27327:
27325:
27319:
27318:
27316:
27315:
27310:
27304:
27302:
27296:
27295:
27293:
27292:
27287:
27282:
27277:
27272:
27267:
27262:
27257:
27252:
27247:
27242:
27237:
27232:
27227:
27222:
27217:
27212:
27207:
27202:
27197:
27192:
27190:Almost perfect
27187:
27181:
27179:
27169:
27168:
27152:
27151:
27148:
27147:
27145:
27144:
27139:
27134:
27129:
27124:
27119:
27114:
27109:
27104:
27099:
27094:
27089:
27083:
27080:
27079:
27067:
27066:
27063:
27062:
27060:
27059:
27054:
27049:
27044:
27038:
27035:
27034:
27022:
27021:
27018:
27017:
27015:
27014:
27009:
27004:
26999:
26997:Stirling first
26994:
26989:
26984:
26979:
26974:
26969:
26964:
26959:
26954:
26949:
26944:
26939:
26934:
26929:
26924:
26919:
26913:
26910:
26909:
26899:
26898:
26895:
26894:
26891:
26890:
26887:
26886:
26884:
26883:
26878:
26873:
26867:
26865:
26858:
26852:
26851:
26848:
26847:
26845:
26844:
26838:
26836:
26830:
26829:
26827:
26826:
26821:
26816:
26811:
26806:
26801:
26795:
26793:
26787:
26786:
26784:
26783:
26778:
26773:
26768:
26763:
26757:
26755:
26746:
26740:
26739:
26736:
26735:
26733:
26732:
26727:
26722:
26717:
26712:
26707:
26702:
26697:
26692:
26687:
26681:
26679:
26673:
26672:
26670:
26669:
26664:
26659:
26654:
26649:
26644:
26639:
26634:
26629:
26623:
26621:
26612:
26602:
26601:
26589:
26588:
26585:
26584:
26582:
26581:
26576:
26571:
26566:
26561:
26556:
26550:
26547:
26546:
26536:
26535:
26532:
26531:
26529:
26528:
26523:
26518:
26513:
26508:
26502:
26499:
26498:
26488:
26487:
26484:
26483:
26481:
26480:
26475:
26470:
26465:
26460:
26455:
26450:
26445:
26439:
26436:
26435:
26422:
26421:
26418:
26417:
26415:
26414:
26409:
26404:
26399:
26394:
26388:
26385:
26384:
26374:
26373:
26370:
26369:
26367:
26366:
26361:
26356:
26351:
26346:
26341:
26336:
26330:
26327:
26326:
26312:
26311:
26308:
26307:
26305:
26304:
26299:
26294:
26289:
26284:
26279:
26274:
26269:
26264:
26259:
26254:
26249:
26244:
26239:
26233:
26230:
26229:
26216:
26215:
26207:
26206:
26199:
26192:
26184:
26178:
26177:
26169:
26151:
26136:
26131:
26125:
26111:
26110:External links
26108:
26107:
26106:
26101:
26088:
26076:
26071:
26063:Broadway Books
26049:
26043:
26030:
26025:
26003:
25997:
25981:
25975:
25962:
25956:
25937:
25934:
25931:
25930:
25928:, p. 193.
25918:
25916:, p. 176.
25906:
25862:
25822:
25806:
25794:
25756:
25747:
25722:
25716:
25681:
25679:, p. 112.
25669:
25643:
25637:
25615:
25598:
25578:
25566:
25564:, p. 110.
25554:
25530:
25512:
25495:
25489:
25471:
25430:
25407:
25391:
25377:
25362:
25347:(in Russian),
25333:
25327:
25306:
25287:
25267:
25241:
25214:
25208:
25190:
25178:
25140:
25115:
25081:
25045:
25010:
24998:
24986:
24974:
24968:
24947:
24922:
24905:
24867:
24837:
24787:
24761:
24709:
24690:
24656:
24650:
24632:
24604:
24572:
24547:
24535:
24483:
24479:
24473:
24470:
24467:
24463:
24445:
24418:
24400:
24394:
24372:
24362:(19): 539â41,
24344:
24315:
24299:
24257:
24227:
24222:978-0521898065
24221:
24201:
24195:
24177:
24165:
24143:
24119:
24084:
24070:
24064:
24046:
24021:
23996:
23984:
23982:, p. 156.
23972:
23966:
23940:
23930:
23910:
23899:
23878:
23857:
23830:
23824:
23806:
23779:
23767:
23736:
23724:
23709:
23695:Academic Press
23675:
23653:
23651:, p. 197.
23638:
23627:
23604:
23602:, p. 180.
23592:
23580:
23568:
23556:
23539:
23513:
23482:
23476:
23451:
23422:
23421:
23419:
23416:
23413:
23412:
23398:
23397:
23395:
23392:
23390:
23387:
23386:
23385:
23379:
23373:
23370:Fibonacci word
23367:
23362:
23354:
23351:
23350:
23349:
23335:
23329:
23322:
23314:
23291:
23281:
23246:
23243:
23224:
23221:
23216:
23212:
23187:
23184:
23179:
23175:
23150:
23147:
23142:
23138:
23113:
23110:
23105:
23101:
23078:
23075:
23070:
23066:
23029:
23017:
23006:
22998:
22997:
22994:
22993:in honeybees).
22893:
22888:
22885:
22882:
22876:
22873:
22866:
22862:
22857:
22854:
22848:
22845:
22706:
22703:
22702:
22701:
22696:
22691:
22684:planning poker
22680:
22654:
22647:
22644:Fibonacci heap
22640:
22633:
22618:
22596:
22565:
22562:
22561:
22560:
22557:conformal maps
22545:
22530:Fibonacci cube
22526:
22513:
22508:
22501:
22498:
22495:
22492:
22488:
22482:
22477:
22473:
22467:
22464:
22461:
22457:
22451:
22448:
22445:
22441:
22437:
22434:
22431:
22426:
22422:
22416:
22413:
22410:
22406:
22400:
22396:
22392:
22377:right triangle
22369:
22354:
22347:
22330:
22316:
22304:
22284:
22272:
22246:
22234:
22219:
22165:
22147:
22135:
22097:
22096:
22093:
22092:
22087:
22082:
22077:
22074:
22073:
22068:
22063:
22058:
22053:
22048:
22032:
22025:
22009:
22006:
22003:
21992:domino tilings
21983:
21925:
21920:
21917:
21912:
21906:
21900:
21895:
21892:
21887:
21881:
21875:
21870:
21867:
21862:
21847:
21846:
21843:
21842:
21837:
21832:
21827:
21824:
21823:
21818:
21813:
21808:
21803:
21800:
21799:
21794:
21767:
21763:
21740:
21736:
21730:
21726:
21720:
21715:
21712:
21709:
21705:
21701:
21698:
21695:
21690:
21686:
21682:
21679:
21676:
21673:
21668:
21665:
21662:
21658:
21654:
21651:
21648:
21643:
21639:
21635:
21632:
21629:
21626:
21621:
21617:
21613:
21610:
21607:
21604:
21601:
21598:
21593:
21589:
21585:
21582:
21579:
21571:
21567:
21563:
21560:
21557:
21554:
21550:
21528:
21522:
21517:
21513:
21510:
21507:
21504:
21501:
21495:
21486:
21481:
21477:
21474:
21471:
21465:
21459:
21456:
21453:
21449:
21445:
21440:
21436:
21403:
21400:
21398:
21395:
21394:
21393:
21386:
21356:Perrin numbers
21348:
21332:
21322:
21314:
21304:
21292:
21282:
21274:
21265:
21255:
21244:
21237:
21207:Main article:
21204:
21201:
21173:Pisano periods
21146:Main article:
21143:
21137:
21104:
21101:
21098:
21095:
21092:
21089:
21084:
21079:
21074:
21070:
21063:
21060:
21057:
21052:
21048:
21041:
21038:
21035:
21030:
21026:
21019:
21016:
21013:
21010:
21007:
21002:
20997:
20992:
20988:
20981:
20978:
20975:
20970:
20966:
20959:
20956:
20953:
20948:
20944:
20937:
20934:
20931:
20926:
20922:
20915:
20912:
20909:
20904:
20900:
20881:
20870:
20847:
20844:
20840:
20837:
20832:
20829:
20826:
20823:
20818:
20811:
20807:
20801:
20788:
20785:
20781:
20778:
20773:
20770:
20767:
20764:
20759:
20752:
20748:
20742:
20723:
20720:
20715:
20711:
20702:
20699:
20694:
20690:
20670:
20667:
20663:
20659:
20656:
20651:
20643:
20640:
20632:
20627:
20623:
20616:
20613:
20606:
20603:
20600:
20596:
20592:
20589:
20584:
20576:
20573:
20565:
20560:
20556:
20549:
20546:
20539:
20536:
20533:
20530:
20525:
20517:
20514:
20506:
20465:
20462:
20458:
20455:
20450:
20447:
20444:
20441:
20436:
20429:
20425:
20419:
20406:
20403:
20399:
20396:
20391:
20388:
20385:
20382:
20379:
20374:
20367:
20363:
20357:
20338:
20335:
20330:
20326:
20317:
20314:
20309:
20305:
20285:
20282:
20278:
20274:
20271:
20266:
20258:
20255:
20247:
20242:
20238:
20231:
20228:
20221:
20218:
20215:
20212:
20208:
20204:
20201:
20196:
20188:
20185:
20177:
20172:
20168:
20161:
20158:
20151:
20148:
20145:
20142:
20137:
20129:
20126:
20118:
20077:
20074:
20070:
20067:
20062:
20059:
20056:
20053:
20048:
20041:
20037:
20031:
20018:
20015:
20011:
20008:
20003:
20000:
19997:
19994:
19989:
19982:
19978:
19972:
19953:
19950:
19945:
19941:
19932:
19929:
19924:
19920:
19900:
19897:
19893:
19889:
19886:
19881:
19873:
19870:
19862:
19857:
19853:
19846:
19843:
19836:
19833:
19830:
19826:
19822:
19819:
19814:
19806:
19803:
19795:
19790:
19786:
19779:
19776:
19769:
19766:
19763:
19760:
19755:
19747:
19744:
19736:
19695:
19692:
19688:
19685:
19680:
19677:
19674:
19671:
19668:
19663:
19656:
19652:
19646:
19633:
19630:
19626:
19623:
19618:
19615:
19612:
19609:
19606:
19601:
19594:
19590:
19584:
19565:
19562:
19557:
19553:
19544:
19541:
19536:
19532:
19512:
19509:
19506:
19502:
19498:
19495:
19490:
19482:
19479:
19471:
19466:
19462:
19455:
19452:
19445:
19442:
19439:
19436:
19432:
19428:
19425:
19420:
19412:
19409:
19401:
19396:
19392:
19385:
19382:
19375:
19372:
19369:
19366:
19361:
19353:
19350:
19342:
19300:
19295:
19291:
19288:
19284:
19281:
19276:
19273:
19270:
19262:
19259:
19256:
19252:
19249:
19243:
19239:
19236:
19231:
19223:
19220:
19212:
19207:
19203:
19196:
19193:
19187:
19186:
19182:
19179:
19175:
19172:
19167:
19164:
19161:
19153:
19150:
19147:
19143:
19140:
19134:
19130:
19127:
19122:
19114:
19111:
19103:
19098:
19094:
19087:
19084:
19078:
19077:
19075:
19070:
19065:
19057:
19053:
19050:
19047:
19041:
19035:
18999:
18995:
18990:
18986:
18981:
18978:
18973:
18970:
18964:
18959:
18956:
18951:
18947:
18944:
18940:
18907:
18904:
18901:
18899:
18895:
18891:
18887:
18885:
18882:
18879:
18876:
18874:
18870:
18866:
18862:
18860:
18857:
18854:
18851:
18848:
18846:
18842:
18834:
18831:
18823:
18818:
18817:
18814:
18811:
18808:
18805:
18803:
18799:
18795:
18791:
18789:
18786:
18783:
18780:
18778:
18774:
18770:
18766:
18764:
18761:
18758:
18755:
18752:
18750:
18746:
18738:
18735:
18727:
18722:
18721:
18718:
18715:
18712:
18709:
18707:
18703:
18699:
18695:
18693:
18690:
18687:
18684:
18682:
18678:
18670:
18667:
18659:
18654:
18653:
18650:
18647:
18644:
18641:
18639:
18635:
18631:
18627:
18625:
18622:
18619:
18616:
18614:
18610:
18606:
18602:
18600:
18597:
18594:
18591:
18588:
18586:
18582:
18574:
18571:
18563:
18558:
18557:
18554:
18551:
18548:
18545:
18543:
18539:
18535:
18531:
18529:
18526:
18523:
18520:
18518:
18514:
18510:
18506:
18504:
18501:
18498:
18495:
18492:
18490:
18486:
18478:
18475:
18467:
18462:
18461:
18439:
18435:
18432:
18428:
18425:
18420:
18417:
18411:
18406:
18403:
18398:
18394:
18391:
18387:
18375:
18372:
18368:
18365:
18359:
18354:
18351:
18346:
18342:
18337:
18333:
18304:
18299:
18295:
18292:
18288:
18285:
18280:
18277:
18274:
18271:
18263:
18261:
18258:
18255:
18254:
18250:
18247:
18243:
18240:
18235:
18232:
18229:
18226:
18218:
18216:
18213:
18212:
18209:
18206:
18203:
18195:
18193:
18190:
18189:
18187:
18182:
18178:
18173:
18170:
18165:
18142:
18134:
18131:
18123:
18098:
18089:
18076:
18067:
18058:
18051:
18043:
18042:Prime divisors
18040:
18032:perfect number
17996:
17975:
17964:
17953:
17944:
17943:
17924:Main article:
17921:
17918:
17891:
17887:
17884:
17880:
17877:
17870:
17864:
17858:
17855:
17853:
17850:
17849:
17846:
17843:
17841:
17838:
17837:
17835:
17829:
17824:
17816:
17813:
17810:
17806:
17802:
17798:
17794:
17790:
17789:
17784:
17780:
17776:
17772:
17769:
17766:
17762:
17758:
17757:
17755:
17731:
17679:
17673:
17668:
17665:
17660:
17655:
17651:
17647:
17643:
17640:
17629:primality test
17624:
17621:
17609:
17603:
17598:
17595:
17590:
17585:
17581:
17577:
17573:
17570:
17538:
17533:
17528:
17525:
17522:
17518:
17514:
17511:
17508:
17505:
17502:
17499:
17495:
17492:
17487:
17484:
17481:
17478:
17475:
17474:
17471:
17466:
17463:
17460:
17456:
17452:
17449:
17446:
17443:
17440:
17437:
17433:
17430:
17425:
17422:
17419:
17416:
17413:
17412:
17409:
17404:
17400:
17396:
17393:
17390:
17387:
17385:
17382:
17379:
17376:
17375:
17373:
17356:
17335:
17315:
17279:
17278:
17267:
17264:
17261:
17256:
17253:
17250:
17246:
17242:
17237:
17234:
17231:
17227:
17223:
17220:
17217:
17214:
17209:
17206:
17203:
17199:
17195:
17190:
17186:
17182:
17179:
17176:
17173:
17168:
17165:
17162:
17158:
17154:
17149:
17145:
17141:
17138:
17115:
17112:
17107:
17103:
17082:
17079:
17074:
17070:
17031:
17028:
17025:
17022:
17019:
17016:
17013:
17010:
17007:
17004:
17000:
16996:
16993:
16990:
16987:
16982:
16978:
16974:
16969:
16965:
16961:
16956:
16952:
16948:
16945:
16928:
16910:
16907:
16902:
16898:
16885:
16882:
16880:
16877:
16865:
16856:
16852:
16847:
16841:
16838:
16833:
16829:
16824:
16818:
16815:
16812:
16803:
16799:
16794:
16790:
16783:
16778:
16775:
16772:
16768:
16743:
16738:
16732:
16727:
16724:
16718:
16709:
16705:
16700:
16696:
16689:
16684:
16681:
16678:
16674:
16637:
16634:
16633:3.359885666243
16631:
16624:
16621:
16617:
16613:
16606:
16601:
16598:
16595:
16591:
16587:
16580:
16577:
16574:
16571:
16567:
16563:
16556:
16551:
16548:
16545:
16541:
16537:
16530:
16526:
16522:
16515:
16510:
16507:
16504:
16500:
16472:
16467:
16458:
16455:
16451:
16447:
16444:
16438:
16435:
16431:
16425:
16417:
16414:
16410:
16406:
16403:
16397:
16394:
16390:
16383:
16377:
16372:
16365:
16362:
16358:
16354:
16330:
16322:
16318:
16314:
16311:
16305:
16301:
16293:
16288:
16285:
16282:
16278:
16274:
16271:
16268:
16265:
16262:
16251:Lambert series
16237:
16233:
16228:
16224:
16220:
16217:
16214:
16211:
16206:
16202:
16198:
16195:
16191:
16185:
16180:
16173:
16170:
16166:
16162:
16155:
16150:
16147:
16144:
16140:
16117:
16112:
16108:
16105:
16100:
16092:
16084:
16077:
16073:
16065:
16060:
16057:
16054:
16050:
16042:
16039:
16036:
16032:
16028:
16025:
16022:
16014:
16009:
16006:
16003:
15999:
15968:
15963:
15959:
15953:
15945:
15942:
15939:
15936:
15932:
15928:
15925:
15921:
15914:
15909:
15906:
15903:
15899:
15876:
15872:
15868:
15865:
15860:
15855:
15849:
15843:
15838:
15835:
15829:
15826:
15822:
15814:
15810:
15806:
15801:
15796:
15790:
15784:
15779:
15776:
15770:
15767:
15763:
15755:
15751:
15746:
15739:
15736:
15731:
15724:
15717:
15713:
15707:
15700:
15695:
15692:
15689:
15685:
15662:
15657:
15652:
15646:
15640:
15635:
15632:
15626:
15623:
15619:
15611:
15607:
15600:
15596:
15590:
15583:
15580:
15577:
15574:
15570:
15566:
15559:
15554:
15551:
15548:
15544:
15523:
15520:
15508:
15505:
15502:
15499:
15494:
15491:
15486:
15481:
15478:
15473:
15470:
15467:
15464:
15461:
15441:
15438:
15435:
15432:
15412:
15390:
15386:
15380:
15377:
15372:
15368:
15363:
15360:
15357:
15354:
15351:
15348:
15334:satisfies the
15323:
15320:
15317:
15314:
15289:
15285:
15281:
15277:
15274:
15270:
15266:
15263:
15260:
15257:
15229:
15223:
15218:
15215:
15211:
15204:
15201:
15195:
15192:
15171:
15165:
15160:
15157:
15153:
15146:
15143:
15137:
15134:
15113:
15106:
15103:
15100:
15097:
15093:
15088:
15082:
15079:
15076:
15073:
15069:
15063:
15056:
15052:
15047:
15044:
15041:
15038:
15035:
15008:
15005:
15002:
14980:
14976:
14949:
14946:
14943:
14940:
14938:
14936:
14931:
14927:
14923:
14918:
14915:
14912:
14908:
14904:
14899:
14896:
14893:
14889:
14885:
14880:
14876:
14872:
14867:
14862:
14859:
14856:
14852:
14848:
14843:
14839:
14835:
14832:
14827:
14823:
14819:
14816:
14811:
14807:
14803:
14800:
14797:
14795:
14793:
14788:
14784:
14778:
14775:
14772:
14768:
14762:
14757:
14754:
14751:
14747:
14743:
14738:
14734:
14728:
14725:
14722:
14718:
14712:
14707:
14704:
14701:
14697:
14693:
14688:
14684:
14678:
14674:
14668:
14663:
14660:
14657:
14653:
14649:
14646:
14644:
14642:
14637:
14634:
14631:
14627:
14621:
14617:
14611:
14606:
14603:
14600:
14596:
14592:
14587:
14584:
14581:
14577:
14571:
14567:
14561:
14556:
14553:
14550:
14546:
14542:
14537:
14533:
14527:
14523:
14517:
14512:
14509:
14506:
14502:
14498:
14495:
14493:
14491:
14488:
14485:
14482:
14479:
14474:
14470:
14466:
14463:
14460:
14457:
14454:
14451:
14450:
14430:
14425:
14421:
14417:
14414:
14411:
14408:
14405:
14383:
14375:
14371:
14367:
14364:
14361:
14358:
14354:
14349:
14346:
14343:
14340:
14337:
14315:
14312:
14308:
14304:
14301:
14297:
14293:
14289:
14268:
14258:complex number
14243:
14240:
14237:
14232:
14228:
14224:
14221:
14216:
14212:
14208:
14205:
14200:
14196:
14192:
14189:
14184:
14180:
14176:
14173:
14170:
14167:
14164:
14159:
14155:
14149:
14145:
14139:
14134:
14131:
14128:
14124:
14120:
14117:
14114:
14111:
14108:
14087:
14084:
14058:
14053:
14050:
14047:
14040:
14037:
14034:
14030:
14022:
14015:
14011:
14003:
14000:
13997:
13993:
13986:
13981:
13978:
13973:
13965:
13960:
13957:
13954:
13950:
13946:
13941:
13938:
13935:
13932:
13928:
13903:
13898:
13895:
13892:
13885:
13882:
13879:
13875:
13867:
13860:
13856:
13848:
13845:
13842:
13838:
13831:
13826:
13823:
13818:
13810:
13805:
13802:
13799:
13795:
13791:
13786:
13783:
13780:
13777:
13773:
13734:
13728:
13721:
13718:
13715:
13711:
13705:
13702:
13697:
13690:
13686:
13679:
13673:
13666:
13662:
13656:
13653:
13649:
13643:
13636:
13632:
13626:
13623:
13618:
13611:
13608:
13605:
13601:
13594:
13588:
13585:
13582:
13578:
13572:
13568:
13564:
13561:
13556:
13553:
13549:
13527:
13520:
13516:
13510:
13505:
13501:
13495:
13488:
13485:
13482:
13478:
13472:
13469:
13464:
13457:
13454:
13451:
13447:
13441:
13436:
13433:
13430:
13427:
13423:
13401:
13394:
13390:
13384:
13379:
13372:
13368:
13360:
13357:
13354:
13350:
13346:
13343:
13338:
13331:
13328:
13325:
13321:
13315:
13310:
13307:
13304:
13301:
13297:
13272:
13268:
13262:
13258:
13254:
13251:
13248:
13245:
13242:
13237:
13230:
13226:
13220:
13217:
13212:
13209:
13206:
13202:
13196:
13193:
13190:
13186:
13180:
13176:
13172:
13169:
13164:
13157:
13153:
13147:
13144:
13139:
13136:
13132:
13104:
13086:
13082:
13076:
13072:
13068:
13064:
13058:
13055:
13052:
13048:
13044:
13039:
13036:
13033:
13029:
13024:
13018:
13014:
13010:
13005:
12998:
12995:
12992:
12988:
12982:
12977:
12970:
12967:
12964:
12960:
12954:
12949:
12946:
12942:
12920:
12917:
12914:
12910:
12904:
12900:
12896:
12893:
12890:
12887:
12882:
12878:
12872:
12869:
12866:
12862:
12858:
12853:
12850:
12847:
12843:
12837:
12833:
12821:
12818:
12804:
12797:
12793:
12785:
12782:
12779:
12775:
12771:
12768:
12765:
12762:
12757:
12754:
12751:
12747:
12741:
12738:
12735:
12731:
12727:
12722:
12715:
12711:
12686:
12683:
12680:
12676:
12672:
12669:
12666:
12663:
12658:
12655:
12652:
12648:
12642:
12639:
12636:
12632:
12628:
12623:
12616:
12612:
12594:Main article:
12591:
12588:
12583:
12580:
12565:
12560:
12557:
12554:
12551:
12531:
12528:
12525:
12522:
12519:
12495:
12492:
12489:
12484:
12481:
12478:
12474:
12470:
12465:
12460:
12457:
12455:
12453:
12450:
12447:
12444:
12441:
12438:
12435:
12430:
12427:
12424:
12420:
12416:
12411:
12408:
12405:
12401:
12397:
12394:
12392:
12390:
12384:
12381:
12376:
12373:
12368:
12365:
12362:
12358:
12354:
12351:
12348:
12343:
12340:
12337:
12333:
12329:
12323:
12320:
12318:
12316:
12310:
12307:
12302:
12299:
12294:
12291:
12288:
12284:
12277:
12271:
12268:
12263:
12260:
12255:
12252:
12249:
12245:
12238:
12235:
12233:
12231:
12225:
12222:
12219:
12214:
12211:
12206:
12203:
12200:
12196:
12189:
12183:
12180:
12177:
12172:
12169:
12164:
12161:
12158:
12154:
12147:
12144:
12142:
12140:
12136:
12130:
12126:
12122:
12119:
12116:
12111:
12107:
12103:
12100:
12097:
12094:
12090:
12086:
12081:
12077:
12073:
12070:
12067:
12062:
12058:
12054:
12051:
12048:
12046:
12043:
12040:
12039:
12017:
11995:
11992:
11987:
11984:
11981:
11977:
11973:
11968:
11964:
11958:
11953:
11950:
11947:
11943:
11919:
11914:
11910:
11906:
11901:
11897:
11893:
11888:
11884:
11878:
11864:
11861:
11847:
11844:
11841:
11837:
11831:
11828:
11825:
11821:
11817:
11812:
11807:
11803:
11797:
11794:
11791:
11786:
11783:
11780:
11776:
11755:
11749:
11746:
11743:
11739:
11735:
11730:
11726:
11721:
11715:
11712:
11709:
11705:
11701:
11696:
11689:
11686:
11683:
11679:
11673:
11668:
11663:
11659:
11653:
11648:
11645:
11642:
11638:
11615:
11612:
11609:
11605:
11599:
11595:
11591:
11586:
11581:
11577:
11571:
11566:
11563:
11560:
11556:
11533:
11526:
11523:
11520:
11516:
11491:
11488:
11483:
11480:
11477:
11473:
11469:
11464:
11460:
11454:
11451:
11448:
11443:
11440:
11437:
11433:
11413:
11410:
11407:
11396:
11395:
11384:
11381:
11376:
11373:
11370:
11366:
11362:
11357:
11354:
11351:
11347:
11343:
11338:
11335:
11332:
11328:
11324:
11319:
11315:
11309:
11304:
11301:
11298:
11294:
11268:
11265:
11262:
11258:
11237:
11234:
11229:
11226:
11223:
11219:
11215:
11210:
11206:
11200:
11195:
11192:
11189:
11185:
11165:
11162:
11150:
11142:
11138:
11134:
11131:
11128:
11125:
11121:
11094:
11090:
11084:
11080:
11074:
11069:
11066:
11063:
11059:
11031:
11028:
11025:
10997:
10977:
10970:
10966:
10962:
10959:
10954:
10951:
10948:
10917:
10913:
10910:
10906:
10900:
10896:
10892:
10880:
10877:
10851:
10847:
10843:
10840:
10837:
10834:
10831:
10826:
10823:
10820:
10816:
10812:
10807:
10803:
10780:
10777:
10774:
10770:
10766:
10761:
10757:
10722:
10718:
10695:
10692:
10689:
10685:
10679:
10675:
10671:
10666:
10661:
10657:
10651:
10646:
10643:
10640:
10636:
10602:
10599:
10595:
10560:
10557:
10554:
10551:
10547:
10522:
10519:
10514:
10511:
10508:
10505:
10501:
10497:
10492:
10489:
10485:
10479:
10474:
10471:
10468:
10464:
10441:
10438:
10434:
10430:
10425:
10422:
10419:
10416:
10412:
10406:
10403:
10400:
10395:
10392:
10389:
10385:
10361:
10358:
10353:
10350:
10347:
10343:
10339:
10334:
10330:
10324:
10319:
10316:
10313:
10309:
10288:
10285:
10280:
10276:
10264:
10263:
10251:
10247:
10244:
10241:
10238:
10235:
10232:
10229:
10226:
10223:
10220:
10217:
10214:
10211:
10207:
10203:
10199:
10195:
10192:
10189:
10186:
10183:
10180:
10177:
10174:
10171:
10168:
10165:
10162:
10159:
10156:
10153:
10149:
10145:
10142:
10139:
10136:
10133:
10128:
10125:
10122:
10118:
10114:
10109:
10105:
10101:
10096:
10093:
10090:
10086:
10059:
10056:
10053:
10050:
10047:
10044:
10041:
10038:
10035:
10032:
10029:
10026:
10023:
10020:
10017:
10014:
10011:
10008:
10005:
10002:
9999:
9996:
9993:
9990:
9987:
9984:
9981:
9978:
9975:
9955:
9952:
9949:
9946:
9943:
9940:
9937:
9934:
9931:
9928:
9925:
9922:
9919:
9916:
9913:
9910:
9907:
9904:
9901:
9898:
9895:
9892:
9872:
9869:
9866:
9844:
9841:
9836:
9833:
9830:
9826:
9822:
9817:
9813:
9807:
9802:
9799:
9796:
9792:
9754:
9750:
9727:
9724:
9721:
9717:
9713:
9708:
9705:
9702:
9698:
9675:
9672:
9669:
9665:
9642:
9639:
9636:
9632:
9611:
9608:
9605:
9585:
9582:
9579:
9558:
9554:
9551:
9548:
9545:
9542:
9539:
9536:
9533:
9530:
9527:
9524:
9521:
9518:
9515:
9512:
9509:
9506:
9503:
9500:
9497:
9494:
9490:
9486:
9482:
9478:
9475:
9472:
9469:
9466:
9463:
9460:
9457:
9454:
9451:
9448:
9445:
9442:
9439:
9436:
9433:
9430:
9427:
9424:
9421:
9418:
9414:
9410:
9405:
9401:
9378:
9374:
9351:
9348:
9345:
9341:
9337:
9332:
9329:
9326:
9322:
9318:
9313:
9309:
9297:
9296:
9284:
9280:
9277:
9274:
9271:
9268:
9265:
9262:
9259:
9256:
9253:
9250:
9247:
9244:
9241:
9238:
9235:
9232:
9229:
9226:
9223:
9220:
9217:
9214:
9211:
9208:
9205:
9202:
9199:
9196:
9193:
9190:
9187:
9184:
9181:
9178:
9175:
9172:
9169:
9166:
9163:
9160:
9156:
9152:
9149:
9146:
9141:
9137:
9126:
9114:
9110:
9107:
9104:
9101:
9098:
9095:
9092:
9089:
9086:
9083:
9080:
9077:
9074:
9071:
9068:
9065:
9062:
9059:
9056:
9053:
9050:
9046:
9042:
9039:
9036:
9031:
9027:
9016:
9004:
9000:
8997:
8994:
8991:
8988:
8985:
8982:
8979:
8976:
8973:
8970:
8966:
8962:
8959:
8956:
8951:
8947:
8936:
8924:
8920:
8917:
8914:
8911:
8908:
8904:
8900:
8897:
8894:
8889:
8885:
8874:
8862:
8858:
8855:
8852:
8849:
8845:
8841:
8838:
8835:
8830:
8826:
8815:
8803:
8799:
8796:
8792:
8788:
8785:
8782:
8777:
8773:
8740:
8735:
8732:
8729:
8724:
8703:
8700:
8695:
8691:
8670:
8667:
8662:
8658:
8635:
8631:
8610:
8607:
8604:
8582:
8578:
8561:
8558:
8556:
8553:
8511:
8506:
8502:
8498:
8493:
8489:
8485:
8480:
8477:
8474:
8470:
8466:
8463:
8460:
8457:
8455:
8453:
8448:
8444:
8440:
8435:
8431:
8427:
8422:
8419:
8416:
8412:
8408:
8405:
8402:
8399:
8397:
8395:
8390:
8386:
8382:
8377:
8374:
8371:
8367:
8363:
8358:
8355:
8352:
8348:
8344:
8341:
8338:
8336:
8329:
8326:
8318:
8315:
8311:
8307:
8306:
8301:
8294:
8291:
8288:
8284:
8278:
8273:
8266:
8262:
8256:
8253:
8251:
8247:
8244:
8241:
8238:
8234:
8230:
8229:
8193:
8188:
8185:
8182:
8178:
8174:
8171:
8169:
8165:
8161:
8155:
8152:
8149:
8145:
8141:
8136:
8133:
8130:
8126:
8120:
8116:
8112:
8111:
8108:
8103:
8100:
8097:
8094:
8091:
8087:
8083:
8080:
8078:
8073:
8070:
8067:
8063:
8055:
8052:
8049:
8045:
8040:
8034:
8030:
8022:
8018:
8013:
8012:
7990:matrix product
7949:
7944:
7937:
7933:
7927:
7922:
7919:
7916:
7912:
7906:
7903:
7900:
7896:
7892:
7887:
7883:
7879:
7876:
7873:
7825:
7820:
7812:
7809:
7806:
7802:
7798:
7794:
7790:
7786:
7785:
7780:
7776:
7772:
7768:
7765:
7762:
7758:
7754:
7753:
7751:
7746:
7741:
7735:
7729:
7726:
7724:
7721:
7720:
7717:
7714:
7712:
7709:
7708:
7706:
7686:
7676:
7653:
7643:
7634:
7608:
7590:
7587:
7584:
7572:
7561:
7558:
7546:
7535:
7532:
7520:
7509:
7506:
7503:
7500:
7451:
7442:
7427:
7424:
7420:
7416:
7413:
7410:
7407:
7402:
7398:
7386:
7381:
7377:
7350:
7344:
7339:
7336:
7331:
7323:
7317:
7314:
7312:
7309:
7306:
7305:
7300:
7297:
7293:
7289:
7287:
7284:
7283:
7281:
7273:
7269:
7262:
7254:
7251:
7247:
7243:
7240:
7237:
7234:
7232:
7229:
7228:
7225:
7222:
7218:
7214:
7210:
7209:
7207:
7200:
7194:
7191:
7189:
7186:
7185:
7180:
7177:
7173:
7169:
7166:
7164:
7161:
7160:
7158:
7153:
7150:
7148:
7146:
7140:
7133:
7129:
7123:
7119:
7113:
7105:
7102:
7098:
7092:
7084:
7081:
7077:
7073:
7070:
7067:
7064:
7062:
7059:
7058:
7055:
7052:
7048:
7044:
7040:
7039:
7037:
7032:
7029:
7026:
7024:
7022:
7016:
7009:
7005:
6999:
6995:
6989:
6981:
6978:
6974:
6968:
6964:
6960:
6957:
6954:
6952:
6950:
6944:
6937:
6933:
6927:
6923:
6917:
6909:
6905:
6901:
6898:
6896:
6891:
6884:
6880:
6874:
6871:
6868:
6864:
6858:
6852:
6851:
6823:
6818:
6812:
6809:
6807:
6804:
6803:
6798:
6795:
6791:
6787:
6784:
6782:
6779:
6778:
6776:
6771:
6768:
6764:
6759:
6750:
6747:
6743:
6739:
6736:
6734:
6731:
6730:
6727:
6724:
6722:
6719:
6718:
6716:
6711:
6708:
6680:
6675:
6672:
6668:
6662:
6658:
6654:
6651:
6648:
6646:
6642:
6638:
6634:
6633:
6630:
6625:
6622:
6618:
6614:
6611:
6608:
6605:
6603:
6601:
6598:
6597:
6559:
6554:
6548:
6540:
6526:
6521:
6518:
6507:
6496:
6483:
6471:
6466:
6460:
6452:
6438:
6433:
6430:
6419:
6408:
6395:
6384:
6379:
6375:
6340:
6334:
6329:
6323:
6320:
6316:
6312:
6306:
6298:
6292:
6284:
6270:
6265:
6262:
6251:
6240:
6227:
6215:
6208:
6203:
6200:
6195:
6187:
6181:
6173:
6159:
6154:
6151:
6140:
6129:
6116:
6105:
6102:
6100:
6098:
6092:
6089:
6081:
6078:
6074:
6070:
6067:
6064:
6058:
6054:
6049:
6043:
6040:
6032:
6028:
6021:
6017:
6012:
6009:
6007:
6005:
5999:
5996:
5988:
5984:
5977:
5973:
5968:
5962:
5959:
5951:
5947:
5940:
5936:
5931:
5928:
5926:
5922:
5915:
5912:
5905:
5904:
5878:
5872:
5869:
5860:
5856:
5851:
5845:
5842:
5833:
5829:
5824:
5818:
5813:
5810:
5805:
5799:
5794:
5787:
5784:
5758:
5752:
5747:
5741:
5738:
5734:
5730:
5724:
5718:
5712:
5709:
5702:
5696:
5691:
5688:
5683:
5677:
5671:
5668:
5640:
5629:
5624:
5621:
5616:
5608:
5605:
5599:
5594:
5591:
5587:
5583:
5580:
5577:
5555:
5544:
5539:
5536:
5531:
5523:
5520:
5514:
5511:
5476:
5469:
5466:
5457:
5452:
5447:
5442:
5435:
5432:
5406:
5401:
5394:
5391:
5383:
5379:
5374:
5371:
5368:
5361:
5358:
5331:
5324:
5320:
5314:
5311:
5308:
5304:
5298:
5290:
5284:
5281:
5279:
5276:
5275:
5272:
5269:
5267:
5264:
5263:
5261:
5256:
5250:
5243:
5240:
5237:
5233:
5227:
5224:
5221:
5217:
5211:
5190:
5187:
5183:
5182:
5171:
5165:
5161:
5155:
5152:
5149:
5145:
5141:
5138:
5133:
5129:
5125:
5122:
5117:
5113:
5109:
5106:
5103:
5100:
5097:
5092:
5085:
5081:
5075:
5052:
5048:
5044:
5039:
5036:
5033:
5029:
5025:
5022:
5017:
5013:
5009:
5004:
4997:
4993:
4989:
4986:
4981:
4978:
4975:
4971:
4967:
4964:
4959:
4955:
4951:
4946:
4942:
4919:
4914:
4906:
4902:
4898:
4895:
4892:
4889:
4886:
4881:
4874:
4870:
4864:
4859:
4854:
4847:
4843:
4836:
4831:
4827:
4798:
4794:
4767:
4763:
4757:
4733:
4727:
4723:
4718:
4715:
4711:
4705:
4701:
4697:
4694:
4691:
4688:
4683:
4679:
4675:
4672:
4667:
4663:
4651:perfect square
4638:
4635:
4630:
4626:
4622:
4602:
4599:
4594:
4590:
4586:
4575:if and only if
4566:
4565:Identification
4563:
4551:
4546:
4543:
4540:
4536:
4532:
4527:
4524:
4521:
4517:
4513:
4508:
4504:
4486:
4463:
4458:
4455:
4452:
4448:
4444:
4441:
4436:
4432:
4428:
4423:
4419:
4398:
4395:
4392:
4389:
4384:
4380:
4359:
4355:
4351:
4348:
4345:
4342:
4322:
4317:
4313:
4309:
4306:
4301:
4298:
4295:
4291:
4287:
4282:
4278:
4274:
4271:
4268:
4263:
4260:
4257:
4253:
4249:
4244:
4240:
4236:
4233:
4230:
4225:
4222:
4219:
4215:
4211:
4208:
4205:
4202:
4199:
4196:
4191:
4187:
4183:
4180:
4175:
4172:
4169:
4165:
4161:
4156:
4152:
4146:
4142:
4138:
4135:
4132:
4127:
4124:
4121:
4117:
4113:
4110:
4105:
4101:
4097:
4094:
4089:
4086:
4083:
4079:
4043:
4038:
4035:
4032:
4028:
4024:
4021:
4016:
4012:
4008:
4003:
3999:
3970:
3948:
3944:
3921:
3918:
3915:
3912:
3909:
3904:
3900:
3887:
3884:
3883:
3882:
3859:
3837:
3833:
3829:
3822:
3818:
3812:
3809:
3806:
3802:
3794:
3791:
3788:
3784:
3756:
3752:
3746:
3742:
3738:
3735:
3730:
3726:
3703:
3699:
3676:
3672:
3649:
3646:
3643:
3636:
3632:
3626:
3623:
3620:
3616:
3608:
3605:
3602:
3598:
3578:
3575:
3556:
3553:
3541:
3535:
3532:
3529:
3524:
3521:
3518:
3515:
3509:
3506:
3503:
3498:
3494:
3490:
3475:
3438:
3434:
3431:
3428:
3425:
3420:
3416:
3412:
3397:
3379:
3373:
3367:
3363:
3345:
3339:
3336:
3323:
3320:
3317:
3314:
3311:
3308:
3305:
3300:
3296:
3275:
3272:
3269:
3266:
3263:
3260:
3257:
3254:
3234:
3231:
3228:
3225:
3220:
3216:
3211:
3207:
3204:
3201:
3198:
3193:
3189:
3185:
3182:
3179:
3176:
3173:
3170:
3166:
3162:
3159:
3156:
3153:
3150:
3147:
3144:
3141:
3138:
3135:
3130:
3126:
3105:
3102:
3099:
3096:
3090:
3086:
3082:
3079:
3075:
3071:
3068:
3065:
3062:
3057:
3052:
3047:
3043:
3038:
3034:
3031:
3028:
3025:
3019:
3016:
3013:
3010:
3007:
3004:
3001:
2996:
2980:floor function
2965:
2962:
2959:
2953:
2949:
2945:
2940:
2935:
2930:
2926:
2921:
2917:
2914:
2911:
2908:
2905:
2861:
2858:
2855:
2849:
2845:
2839:
2833:
2829:
2823:
2819:
2814:
2810:
2782:
2776:
2772:
2750:
2725:
2722:
2717:
2713:
2707:
2701:
2697:
2691:
2678:
2675:
2659:
2653:
2646:
2642:
2638:
2635:
2630:
2626:
2619:
2616:
2614:
2612:
2609:
2608:
2605:
2599:
2594:
2589:
2585:
2581:
2576:
2572:
2565:
2562:
2560:
2558:
2555:
2554:
2528:
2524:
2520:
2517:
2512:
2508:
2504:
2501:
2496:
2492:
2476:
2467:
2446:
2443:
2440:
2437:
2434:
2430:
2424:
2420:
2415:
2409:
2406:
2403:
2399:
2394:
2391:
2366:
2360:
2357:
2354:
2352:
2350:
2347:
2344:
2341:
2338:
2335:
2334:
2331:
2328:
2325:
2323:
2321:
2318:
2315:
2312:
2311:
2308:
2282:
2272:
2262:
2232:
2227:
2224:
2221:
2217:
2213:
2208:
2205:
2202:
2198:
2194:
2191:
2189:
2187:
2182:
2179:
2176:
2172:
2168:
2165:
2160:
2157:
2154:
2150:
2146:
2143:
2138:
2135:
2132:
2128:
2124:
2121:
2116:
2113:
2110:
2106:
2102:
2099:
2096:
2094:
2092:
2089:
2084:
2081:
2078:
2074:
2070:
2065:
2062:
2059:
2055:
2051:
2048:
2045:
2042:
2037:
2034:
2031:
2027:
2023:
2018:
2015:
2012:
2008:
2004:
2001:
1998:
1995:
1993:
1991:
1986:
1982:
1978:
1975:
1970:
1966:
1962:
1959:
1956:
1954:
1950:
1946:
1942:
1941:
1915:
1911:
1907:
1904:
1899:
1895:
1891:
1888:
1883:
1879:
1842:
1837:
1834:
1831:
1827:
1823:
1818:
1815:
1812:
1808:
1804:
1801:
1799:
1795:
1791:
1787:
1786:
1783:
1778:
1775:
1772:
1768:
1764:
1759:
1756:
1753:
1749:
1745:
1742:
1740:
1736:
1732:
1728:
1727:
1697:
1692:
1689:
1686:
1682:
1678:
1673:
1670:
1667:
1663:
1659:
1654:
1650:
1629:
1626:
1623:
1620:
1615:
1611:
1580:
1574:
1571:
1568:
1565:
1558:
1555:
1551:
1547:
1544:
1541:
1538:
1533:
1529:
1522:
1516:
1509:
1506:
1502:
1498:
1495:
1492:
1489:
1484:
1480:
1473:
1468:
1464:
1439:
1436:
1432:
1428:
1425:
1422:
1400:
1397:
1394:
1390:
1387:
1384:
1379:
1376:
1371:
1368:
1365:
1362:
1359:
1356:
1351:
1345:
1340:
1337:
1331:
1328:
1292:
1289:
1285:
1282:
1277:
1271:
1266:
1263:
1257:
1254:
1230:
1224:
1217:
1213:
1209:
1204:
1200:
1193:
1187:
1184:
1181:
1174:
1170:
1166:
1161:
1157:
1150:
1145:
1141:
1098:
1095:
1090:Main article:
1087:
1084:
1078:
1025:
1024:
1021:
1018:
1015:
952:
949:
911:
900:
888:
857:
832:
823:
814:
796:
793:
791:
788:
787:
786:
783:
782:
779:
776:
773:
770:
767:
764:
761:
758:
755:
752:
749:
746:
743:
740:
737:
734:
731:
728:
725:
721:
720:
716:
710:
706:
700:
696:
690:
686:
680:
676:
670:
666:
660:
656:
650:
646:
640:
636:
630:
626:
620:
616:
610:
606:
600:
596:
590:
586:
580:
576:
570:
566:
560:
556:
550:
546:
540:
536:
530:
526:
509:
482:
479:
476:
472:
468:
463:
460:
457:
453:
449:
444:
440:
419:
416:
413:
408:
404:
400:
395:
391:
370:
367:
362:
358:
325:
322:
319:
315:
311:
306:
303:
300:
296:
292:
287:
283:
262:
259:
256:
251:
247:
242:
239:
236:
231:
227:
195:
192:
166:expresses the
125:data structure
122:Fibonacci heap
75:A tiling with
69:
68:
54:
26:
9:
6:
4:
3:
2:
28930:
28919:
28916:
28914:
28911:
28910:
28908:
28891:
28890:
28886:
28885:
28884:
28881:
28879:
28876:
28874:
28871:
28869:
28866:
28865:
28863:
28859:
28853:
28850:
28848:
28845:
28844:
28842:
28838:
28831:
28830:
28826:
28823:
28822:
28818:
28817:
28815:
28811:
28807:
28800:
28795:
28793:
28788:
28786:
28781:
28780:
28777:
28765:
28757:
28756:
28753:
28747:
28744:
28742:
28739:
28737:
28734:
28732:
28729:
28727:
28724:
28722:
28719:
28718:
28716:
28714:
28710:
28704:
28701:
28699:
28696:
28694:
28691:
28689:
28686:
28684:
28681:
28679:
28676:
28674:
28671:
28669:
28666:
28664:
28663:Taylor series
28661:
28660:
28658:
28654:
28644:
28641:
28639:
28636:
28634:
28631:
28629:
28626:
28624:
28621:
28619:
28616:
28614:
28611:
28609:
28606:
28604:
28601:
28599:
28596:
28595:
28593:
28589:
28583:
28580:
28578:
28575:
28573:
28570:
28568:
28565:
28564:
28562:
28558:
28555:
28551:
28541:
28538:
28536:
28533:
28531:
28528:
28527:
28525:
28521:
28515:
28512:
28510:
28507:
28505:
28502:
28500:
28497:
28496:
28494:
28490:
28487:
28483:
28477:
28474:
28472:
28469:
28467:
28464:
28463:
28461:
28457:
28452:
28436:
28433:
28432:
28431:
28428:
28426:
28423:
28421:
28418:
28416:
28413:
28411:
28408:
28406:
28403:
28401:
28398:
28396:
28393:
28391:
28388:
28386:
28383:
28382:
28380:
28376:
28369:
28363:
28360:
28358:
28355:
28353:
28352:Powers of two
28350:
28348:
28345:
28343:
28340:
28338:
28337:Square number
28335:
28333:
28330:
28328:
28325:
28323:
28320:
28319:
28317:
28313:
28310:
28308:
28304:
28300:
28296:
28289:
28284:
28282:
28277:
28275:
28270:
28269:
28266:
28255:
28244:
28241:
28238:
28235:
28231:
28228:
28226:
28223:
28222:
28221:
28218:
28217:
28212:
28209:
28207:
28204:
28202:
28199:
28197:
28194:
28192:
28189:
28187:
28184:
28182:
28179:
28177:
28174:
28172:
28169:
28167:
28164:
28163:
28162:
28159:
28157:
28154:
28153:
28150:
28146:
28139:
28134:
28132:
28127:
28125:
28120:
28119:
28116:
28104:
28100:
28096:
28095:
28092:
28082:
28079:
28078:
28075:
28070:
28065:
28061:
28051:
28048:
28046:
28043:
28042:
28039:
28034:
28029:
28025:
28015:
28012:
28010:
28007:
28006:
28003:
27998:
27993:
27989:
27979:
27976:
27974:
27971:
27970:
27967:
27963:
27957:
27953:
27943:
27940:
27938:
27935:
27933:
27930:
27929:
27926:
27922:
27917:
27913:
27899:
27896:
27895:
27893:
27889:
27883:
27880:
27878:
27875:
27873:
27872:Polydivisible
27870:
27868:
27865:
27863:
27860:
27858:
27855:
27853:
27850:
27849:
27847:
27843:
27837:
27834:
27832:
27829:
27827:
27824:
27822:
27819:
27817:
27814:
27813:
27811:
27808:
27803:
27797:
27794:
27792:
27789:
27787:
27784:
27782:
27779:
27777:
27774:
27772:
27769:
27767:
27764:
27763:
27761:
27758:
27754:
27746:
27743:
27742:
27741:
27738:
27737:
27735:
27732:
27728:
27716:
27713:
27712:
27711:
27708:
27706:
27703:
27701:
27698:
27696:
27693:
27691:
27688:
27686:
27683:
27681:
27678:
27676:
27673:
27671:
27668:
27667:
27665:
27661:
27655:
27652:
27651:
27649:
27645:
27639:
27636:
27634:
27631:
27630:
27628:
27626:Digit product
27624:
27618:
27615:
27613:
27610:
27608:
27605:
27603:
27600:
27599:
27597:
27595:
27591:
27583:
27580:
27578:
27575:
27574:
27573:
27570:
27569:
27567:
27565:
27560:
27556:
27552:
27547:
27542:
27538:
27528:
27525:
27523:
27520:
27518:
27515:
27513:
27510:
27508:
27505:
27503:
27500:
27498:
27495:
27493:
27490:
27488:
27485:
27483:
27480:
27478:
27475:
27473:
27470:
27468:
27465:
27463:
27462:ErdĆsâNicolas
27460:
27458:
27455:
27453:
27450:
27449:
27446:
27441:
27437:
27431:
27427:
27413:
27410:
27408:
27405:
27404:
27402:
27400:
27396:
27390:
27387:
27385:
27382:
27380:
27377:
27375:
27372:
27371:
27369:
27367:
27363:
27357:
27354:
27352:
27349:
27347:
27344:
27342:
27339:
27337:
27334:
27332:
27329:
27328:
27326:
27324:
27320:
27314:
27311:
27309:
27306:
27305:
27303:
27301:
27297:
27291:
27288:
27286:
27283:
27281:
27280:Superabundant
27278:
27276:
27273:
27271:
27268:
27266:
27263:
27261:
27258:
27256:
27253:
27251:
27248:
27246:
27243:
27241:
27238:
27236:
27233:
27231:
27228:
27226:
27223:
27221:
27218:
27216:
27213:
27211:
27208:
27206:
27203:
27201:
27198:
27196:
27193:
27191:
27188:
27186:
27183:
27182:
27180:
27178:
27174:
27170:
27166:
27162:
27157:
27153:
27143:
27140:
27138:
27135:
27133:
27130:
27128:
27125:
27123:
27120:
27118:
27115:
27113:
27110:
27108:
27105:
27103:
27100:
27098:
27095:
27093:
27090:
27088:
27085:
27084:
27081:
27077:
27072:
27068:
27058:
27055:
27053:
27050:
27048:
27045:
27043:
27040:
27039:
27036:
27032:
27027:
27023:
27013:
27010:
27008:
27005:
27003:
27000:
26998:
26995:
26993:
26990:
26988:
26985:
26983:
26980:
26978:
26975:
26973:
26970:
26968:
26965:
26963:
26960:
26958:
26955:
26953:
26950:
26948:
26945:
26943:
26940:
26938:
26935:
26933:
26930:
26928:
26925:
26923:
26920:
26918:
26915:
26914:
26911:
26904:
26900:
26882:
26879:
26877:
26874:
26872:
26869:
26868:
26866:
26862:
26859:
26857:
26856:4-dimensional
26853:
26843:
26840:
26839:
26837:
26835:
26831:
26825:
26822:
26820:
26817:
26815:
26812:
26810:
26807:
26805:
26802:
26800:
26797:
26796:
26794:
26792:
26788:
26782:
26779:
26777:
26774:
26772:
26769:
26767:
26766:Centered cube
26764:
26762:
26759:
26758:
26756:
26754:
26750:
26747:
26745:
26744:3-dimensional
26741:
26731:
26728:
26726:
26723:
26721:
26718:
26716:
26713:
26711:
26708:
26706:
26703:
26701:
26698:
26696:
26693:
26691:
26688:
26686:
26683:
26682:
26680:
26678:
26674:
26668:
26665:
26663:
26660:
26658:
26655:
26653:
26650:
26648:
26645:
26643:
26640:
26638:
26635:
26633:
26630:
26628:
26625:
26624:
26622:
26620:
26616:
26613:
26611:
26610:2-dimensional
26607:
26603:
26599:
26594:
26590:
26580:
26577:
26575:
26572:
26570:
26567:
26565:
26562:
26560:
26557:
26555:
26554:Nonhypotenuse
26552:
26551:
26548:
26541:
26537:
26527:
26524:
26522:
26519:
26517:
26514:
26512:
26509:
26507:
26504:
26503:
26500:
26493:
26489:
26479:
26476:
26474:
26471:
26469:
26466:
26464:
26461:
26459:
26456:
26454:
26451:
26449:
26446:
26444:
26441:
26440:
26437:
26432:
26427:
26423:
26413:
26410:
26408:
26405:
26403:
26400:
26398:
26395:
26393:
26390:
26389:
26386:
26379:
26375:
26365:
26362:
26360:
26357:
26355:
26352:
26350:
26347:
26345:
26342:
26340:
26337:
26335:
26332:
26331:
26328:
26323:
26317:
26313:
26303:
26300:
26298:
26295:
26293:
26292:Perfect power
26290:
26288:
26285:
26283:
26282:Seventh power
26280:
26278:
26275:
26273:
26270:
26268:
26265:
26263:
26260:
26258:
26255:
26253:
26250:
26248:
26245:
26243:
26240:
26238:
26235:
26234:
26231:
26226:
26221:
26217:
26213:
26205:
26200:
26198:
26193:
26191:
26186:
26185:
26182:
26176:
26170:
26166:
26162:
26161:
26156:
26152:
26150:
26146:
26145:
26140:
26137:
26135:
26132:
26129:
26126:
26123:
26119:
26114:
26113:
26104:
26098:
26094:
26089:
26084:
26083:
26077:
26074:
26072:0-7679-0816-3
26068:
26064:
26060:
26059:
26054:
26050:
26046:
26040:
26036:
26031:
26028:
26022:
26018:
26017:
26013:(July 1998),
26012:
26008:
26004:
26000:
25994:
25990:
25986:
25982:
25978:
25972:
25968:
25963:
25959:
25953:
25949:
25945:
25940:
25939:
25927:
25922:
25915:
25910:
25903:
25899:
25894:
25889:
25885:
25881:
25878:(3): 857â78,
25877:
25873:
25866:
25859:
25855:
25851:
25847:
25843:
25839:
25835:
25834:
25826:
25819:
25818:
25810:
25803:
25798:
25785:on 2020-09-25
25781:
25777:
25770:
25763:
25761:
25751:
25743:
25739:
25732:
25726:
25719:
25713:
25709:
25705:
25704:
25699:
25695:
25691:
25685:
25678:
25673:
25666:
25662:
25658:
25654:
25647:
25640:
25634:
25630:
25626:
25619:
25612:
25601:
25599:9782705678128
25595:
25591:
25590:
25582:
25575:
25570:
25563:
25558:
25550:
25543:
25542:
25534:
25526:
25522:
25516:
25508:
25507:
25499:
25492:
25486:
25482:
25475:
25468:on 2006-05-26
25464:
25460:
25456:
25453:(3): 255â74,
25452:
25448:
25441:
25434:
25420:
25419:
25411:
25404:
25403:
25395:
25387:
25381:
25373:
25366:
25359:
25355:
25350:
25346:
25345:
25337:
25330:
25324:
25320:
25316:
25310:
25303:
25298:
25294:
25290:
25284:
25280:
25279:
25271:
25264:
25260:
25256:
25252:
25245:
25237:
25233:
25229:
25227:
25218:
25211:
25205:
25201:
25194:
25187:
25182:
25175:
25171:
25167:
25163:
25159:
25144:
25136:
25135:
25129:
25125:
25119:
25112:
25108:
25104:
25100:
25097:(3): 278â79,
25096:
25092:
25085:
25076:
25068:
25064:
25057:
25056:
25049:
25040:
25033:
25029:
25022:
25021:
25014:
25007:
25002:
24995:
24990:
24983:
24978:
24971:
24965:
24961:
24957:
24951:
24943:
24942:
24936:
24932:
24926:
24918:
24917:
24909:
24902:
24898:
24894:
24890:
24886:
24882:
24878:
24871:
24864:
24860:
24856:
24852:
24848:
24841:
24834:
24830:
24826:
24822:
24818:
24814:
24810:
24806:
24803:(2): 313â18,
24802:
24798:
24791:
24783:
24779:
24772:
24765:
24758:
24754:
24750:
24746:
24742:
24738:
24733:
24728:
24724:
24720:
24713:
24705:
24701:
24694:
24687:
24683:
24679:
24675:
24671:
24667:
24660:
24653:
24647:
24643:
24636:
24623:on 2023-11-18
24619:
24615:
24611:
24607:
24601:
24597:
24593:
24592:Butler, Steve
24586:
24582:
24576:
24568:
24567:
24561:
24557:
24551:
24544:
24543:Prime Numbers
24539:
24531:
24527:
24522:
24517:
24512:
24507:
24504:(3): 366â70,
24503:
24499:
24498:
24481:
24477:
24471:
24468:
24465:
24461:
24449:
24436:on 2009-12-14
24435:
24431:
24430:
24422:
24414:
24410:
24404:
24397:
24395:9781470457181
24391:
24387:
24383:
24376:
24369:
24365:
24361:
24357:
24356:
24348:
24340:
24339:
24333:
24325:
24319:
24312:
24308:
24303:
24296:
24292:
24288:
24284:
24280:
24276:
24272:
24268:
24261:
24253:
24252:
24247:
24243:
24236:
24234:
24232:
24224:
24218:
24214:
24213:
24205:
24198:
24192:
24188:
24181:
24174:
24169:
24159:
24158:
24153:
24147:
24133:
24129:
24123:
24108:
24104:
24103:
24095:
24088:
24080:
24074:
24067:
24061:
24057:
24050:
24042:
24041:
24035:
24031:
24025:
24017:
24016:
24010:
24006:
24000:
23993:
23988:
23981:
23976:
23969:
23963:
23959:
23955:
23951:
23944:
23938:
23933:
23927:
23923:
23922:
23914:
23907:
23902:
23896:
23892:
23888:
23882:
23874:
23870:
23869:
23861:
23847:
23846:
23841:
23834:
23827:
23825:1-4027-3522-7
23821:
23817:
23810:
23795:
23794:
23789:
23783:
23776:
23771:
23756:
23755:
23747:
23740:
23733:
23728:
23720:
23713:
23705:
23700:
23696:
23692:
23688:
23687:
23679:
23672:
23667:
23663:
23657:
23650:
23645:
23643:
23635:
23630:
23624:
23620:
23619:
23614:
23613:Knuth, Donald
23608:
23601:
23596:
23589:
23584:
23577:
23572:
23565:
23560:
23553:
23549:
23542:
23536:
23532:
23531:
23526:
23525:Knuth, Donald
23520:
23518:
23509:
23504:
23501:(3): 229â44,
23500:
23496:
23489:
23487:
23479:
23473:
23469:
23468:
23460:
23458:
23456:
23447:
23446:
23440:
23436:
23430:
23428:
23423:
23409:
23403:
23399:
23383:
23382:Wythoff array
23380:
23377:
23374:
23371:
23368:
23366:
23363:
23360:
23357:
23356:
23347:
23343:
23339:
23336:
23333:
23330:
23327:
23323:
23319:
23315:
23309:
23305:
23301:
23296:
23292:
23289:
23285:
23282:
23277:
23266:
23257:
23253:
23249:
23248:
23242:
23240:
23222:
23219:
23214:
23210:
23185:
23182:
23177:
23173:
23148:
23145:
23140:
23136:
23111:
23108:
23103:
23099:
23076:
23073:
23068:
23064:
23055:
23051:
23041:
23037:
23032:
23028:
23020:
23016:
23009:
23005:
22995:
22992:
22988:
22987:
22986:
22984:
22983:haplodiploids
22980:
22975:
22963:
22959:
22955:
22951:are those at
22944:
22940:
22936:
22932:
22927:
22923:
22919:
22906:
22891:
22886:
22883:
22880:
22874:
22871:
22864:
22860:
22855:
22852:
22846:
22843:
22835:
22831:
22826:
22822:
22818:
22809:
22803:
22799:
22797:
22793:
22789:
22785:
22783:
22779:
22775:
22771:
22767:
22763:
22759:
22755:
22751:
22747:
22743:
22739:
22735:
22727:
22723:
22718:
22712:
22700:
22697:
22695:
22692:
22689:
22685:
22681:
22678:
22674:
22670:
22666:
22663:
22659:
22655:
22652:
22648:
22645:
22641:
22638:
22634:
22631:
22627:
22623:
22619:
22616:
22615:
22610:
22601:
22597:
22594:
22590:
22586:
22582:
22581:
22575:
22570:
22558:
22554:
22550:
22546:
22543:
22539:
22535:
22531:
22527:
22511:
22506:
22499:
22496:
22493:
22490:
22486:
22480:
22475:
22465:
22462:
22459:
22455:
22449:
22446:
22443:
22439:
22435:
22429:
22424:
22414:
22411:
22408:
22404:
22398:
22394:
22382:
22378:
22374:
22370:
22367:
22363:
22359:
22355:
22352:
22348:
22345:
22342:
22338:
22334:
22331:
22315:
22307:
22303:
22285:
22275:
22271:
22264:
22245:
22237:
22233:
22220:
22215:
22210:
22201:
22197:
22193:
22181:
22168:
22164:
22159:
22146:
22138:
22134:
22121:
22120:
22119:
22117:
22113:
22110:
22105:
22103:
22088:
22083:
22078:
22076:
22075:
22069:
22064:
22059:
22054:
22049:
22044:
22043:
22040:
22039:
22038:
22031:
22024:
22007:
22004:
22001:
21993:
21986:
21982:
21978:); there are
21977:
21960:
21956:
21952:
21948:
21918:
21915:
21904:
21893:
21890:
21879:
21868:
21865:
21838:
21833:
21828:
21826:
21825:
21819:
21814:
21809:
21804:
21802:
21801:
21795:
21790:
21789:
21786:
21785:
21784:
21781:
21765:
21761:
21738:
21734:
21728:
21724:
21713:
21710:
21707:
21699:
21696:
21693:
21688:
21680:
21677:
21674:
21666:
21663:
21660:
21656:
21652:
21649:
21646:
21641:
21633:
21630:
21627:
21619:
21615:
21611:
21605:
21602:
21599:
21591:
21587:
21583:
21580:
21577:
21569:
21565:
21561:
21558:
21555:
21552:
21548:
21526:
21515:
21511:
21508:
21505:
21502:
21499:
21484:
21479:
21475:
21472:
21469:
21463:
21457:
21454:
21451:
21447:
21443:
21438:
21434:
21425:
21421:
21413:
21408:
21391:
21387:
21382:
21378:
21374:
21370:
21366:
21362:
21357:
21353:
21349:
21346:
21335:
21331:
21325:
21321:
21317:
21309:
21305:
21302:
21295:
21291:
21285:
21281:
21277:
21264:
21254:
21249:
21248:Lucas numbers
21245:
21242:
21238:
21235:
21234:negafibonacci
21231:
21230:
21229:
21226:
21224:
21220:
21216:
21210:
21200:
21198:
21190:
21186:
21182:
21178:
21174:
21165:
21159:
21149:
21148:Pisano period
21142:
21136:
21132:
21125:
21121:
21115:
21102:
21099:
21096:
21093:
21090:
21087:
21082:
21077:
21072:
21068:
21061:
21058:
21055:
21050:
21046:
21039:
21036:
21033:
21028:
21024:
21017:
21014:
21011:
21008:
21005:
21000:
20995:
20990:
20986:
20979:
20976:
20973:
20968:
20964:
20957:
20954:
20951:
20946:
20942:
20935:
20932:
20929:
20924:
20920:
20913:
20910:
20907:
20902:
20898:
20890:For example,
20888:
20884:
20880:
20873:
20869:
20859:
20842:
20838:
20830:
20827:
20824:
20821:
20816:
20809:
20805:
20799:
20783:
20779:
20771:
20768:
20765:
20762:
20757:
20750:
20746:
20740:
20721:
20718:
20713:
20709:
20700:
20697:
20692:
20688:
20668:
20665:
20661:
20657:
20654:
20641:
20638:
20625:
20621:
20614:
20611:
20604:
20601:
20598:
20594:
20590:
20587:
20574:
20571:
20558:
20554:
20547:
20544:
20537:
20534:
20531:
20528:
20515:
20512:
20495:and we have:
20492:
20485:
20481:
20477:
20460:
20456:
20448:
20445:
20442:
20439:
20434:
20427:
20423:
20417:
20401:
20397:
20389:
20386:
20383:
20380:
20377:
20372:
20365:
20361:
20355:
20336:
20333:
20328:
20324:
20315:
20312:
20307:
20303:
20283:
20280:
20276:
20272:
20269:
20256:
20253:
20240:
20236:
20229:
20226:
20219:
20216:
20213:
20210:
20206:
20202:
20199:
20186:
20183:
20170:
20166:
20159:
20156:
20149:
20146:
20143:
20140:
20127:
20124:
20107:and we have:
20104:
20097:
20093:
20089:
20072:
20068:
20060:
20057:
20054:
20051:
20046:
20039:
20035:
20029:
20013:
20009:
20001:
19998:
19995:
19992:
19987:
19980:
19976:
19970:
19951:
19948:
19943:
19939:
19930:
19927:
19922:
19918:
19898:
19895:
19891:
19887:
19884:
19871:
19868:
19855:
19851:
19844:
19841:
19834:
19831:
19828:
19824:
19820:
19817:
19804:
19801:
19788:
19784:
19777:
19774:
19767:
19764:
19761:
19758:
19745:
19742:
19725:and we have:
19722:
19715:
19711:
19707:
19690:
19686:
19678:
19675:
19672:
19669:
19666:
19661:
19654:
19650:
19644:
19628:
19624:
19616:
19613:
19610:
19607:
19604:
19599:
19592:
19588:
19582:
19563:
19560:
19555:
19551:
19542:
19539:
19534:
19530:
19510:
19507:
19504:
19500:
19496:
19493:
19480:
19477:
19464:
19460:
19453:
19450:
19443:
19440:
19437:
19434:
19430:
19426:
19423:
19410:
19407:
19394:
19390:
19383:
19380:
19373:
19370:
19367:
19364:
19351:
19348:
19331:and we have:
19328:
19321:
19317:
19313:
19293:
19286:
19282:
19274:
19271:
19268:
19254:
19250:
19241:
19237:
19234:
19221:
19218:
19205:
19201:
19194:
19191:
19177:
19173:
19165:
19162:
19159:
19145:
19141:
19132:
19128:
19125:
19112:
19109:
19096:
19092:
19085:
19082:
19073:
19068:
19063:
19055:
19051:
19048:
19045:
19039:
19033:
19023:
19017:
19015:
19010:
18997:
18988:
18984:
18979:
18971:
18968:
18962:
18957:
18954:
18949:
18945:
18942:
18938:
18929:
18922:
18905:
18902:
18900:
18893:
18889:
18883:
18880:
18877:
18875:
18868:
18864:
18858:
18855:
18852:
18849:
18847:
18832:
18829:
18812:
18809:
18806:
18804:
18797:
18793:
18787:
18784:
18781:
18779:
18772:
18768:
18762:
18759:
18756:
18753:
18751:
18736:
18733:
18716:
18713:
18710:
18708:
18701:
18697:
18691:
18688:
18685:
18683:
18668:
18665:
18648:
18645:
18642:
18640:
18633:
18629:
18623:
18620:
18617:
18615:
18608:
18604:
18598:
18595:
18592:
18589:
18587:
18572:
18569:
18552:
18549:
18546:
18544:
18537:
18533:
18527:
18524:
18521:
18519:
18512:
18508:
18502:
18499:
18496:
18493:
18491:
18476:
18473:
18452:For example,
18450:
18437:
18430:
18426:
18418:
18415:
18409:
18404:
18401:
18396:
18392:
18389:
18385:
18370:
18366:
18357:
18352:
18349:
18344:
18340:
18335:
18331:
18317:
18297:
18290:
18286:
18278:
18275:
18272:
18269:
18259:
18256:
18245:
18241:
18233:
18230:
18227:
18224:
18214:
18207:
18204:
18201:
18191:
18185:
18180:
18176:
18171:
18168:
18163:
18132:
18129:
18112:
18103:
18097:
18088:
18083:
18075:
18066:
18057:
18050:
18039:
18037:
18033:
18028:
18026:
18022:
18018:
18013:
18011:
18010:perfect power
18007:
18002:
17995:
17989:
17987:
17983:
17974:
17967:
17963:
17956:
17952:
17947:
17941:
17940:
17939:
17937:
17933:
17927:
17917:
17915:
17911:
17906:
17889:
17882:
17878:
17868:
17862:
17856:
17851:
17844:
17839:
17833:
17827:
17822:
17814:
17811:
17808:
17804:
17796:
17792:
17782:
17778:
17770:
17767:
17764:
17760:
17753:
17743:
17739:
17734:
17730:
17725:
17717:
17709:
17693:
17692:Jacobi symbol
17677:
17671:
17666:
17663:
17658:
17653:
17649:
17645:
17641:
17638:
17630:
17620:
17607:
17601:
17596:
17593:
17588:
17583:
17579:
17575:
17571:
17568:
17560:
17556:
17551:
17531:
17526:
17523:
17520:
17516:
17512:
17509:
17497:
17493:
17485:
17482:
17479:
17476:
17469:
17464:
17461:
17458:
17454:
17450:
17447:
17435:
17431:
17423:
17420:
17417:
17414:
17407:
17402:
17398:
17394:
17391:
17383:
17380:
17377:
17371:
17361:
17359:
17346:
17338:
17334:
17318:
17314:
17301:
17291:
17286:
17265:
17262:
17254:
17251:
17248:
17244:
17240:
17235:
17232:
17229:
17225:
17215:
17207:
17204:
17201:
17197:
17193:
17188:
17184:
17174:
17166:
17163:
17160:
17156:
17152:
17147:
17143:
17129:
17128:
17127:
17113:
17110:
17105:
17101:
17080:
17077:
17072:
17068:
17060:because both
17059:
17054:
17052:
17026:
17023:
17020:
17017:
17014:
17011:
17008:
16998:
16994:
16988:
16985:
16980:
16976:
16972:
16967:
16963:
16959:
16954:
16950:
16935:
16931:
16908:
16905:
16900:
16896:
16876:
16863:
16854:
16850:
16845:
16839:
16836:
16831:
16827:
16822:
16816:
16813:
16810:
16801:
16797:
16792:
16788:
16781:
16776:
16773:
16770:
16766:
16741:
16736:
16730:
16725:
16722:
16716:
16707:
16703:
16698:
16694:
16682:
16679:
16676:
16672:
16663:
16659:
16657:
16653:
16648:
16635:
16632:
16629:
16622:
16619:
16615:
16611:
16599:
16596:
16593:
16589:
16585:
16578:
16575:
16572:
16569:
16565:
16561:
16549:
16546:
16543:
16539:
16535:
16528:
16524:
16520:
16508:
16505:
16502:
16498:
16489:
16484:
16470:
16465:
16456:
16453:
16449:
16445:
16442:
16436:
16433:
16429:
16423:
16415:
16412:
16408:
16404:
16401:
16395:
16392:
16388:
16381:
16375:
16370:
16363:
16360:
16356:
16352:
16328:
16320:
16316:
16312:
16309:
16303:
16299:
16286:
16283:
16280:
16276:
16272:
16266:
16260:
16252:
16235:
16226:
16222:
16215:
16212:
16204:
16200:
16193:
16189:
16183:
16178:
16171:
16168:
16164:
16160:
16148:
16145:
16142:
16138:
16128:
16115:
16110:
16106:
16103:
16098:
16090:
16082:
16075:
16071:
16063:
16058:
16055:
16052:
16048:
16040:
16037:
16034:
16026:
16023:
16007:
16004:
16001:
15997:
15988:
15984:
15979:
15966:
15961:
15957:
15951:
15943:
15940:
15937:
15934:
15930:
15926:
15923:
15919:
15907:
15904:
15901:
15897:
15887:
15874:
15870:
15866:
15863:
15858:
15853:
15847:
15841:
15836:
15833:
15827:
15824:
15820:
15812:
15808:
15804:
15799:
15794:
15788:
15782:
15777:
15774:
15768:
15765:
15761:
15753:
15749:
15744:
15737:
15734:
15729:
15722:
15715:
15711:
15705:
15693:
15690:
15687:
15683:
15673:
15660:
15655:
15650:
15644:
15638:
15633:
15630:
15624:
15621:
15617:
15609:
15605:
15598:
15594:
15588:
15581:
15578:
15575:
15572:
15568:
15564:
15552:
15549:
15546:
15542:
15533:
15529:
15519:
15506:
15503:
15500:
15497:
15492:
15489:
15484:
15479:
15476:
15471:
15465:
15459:
15436:
15430:
15410:
15401:
15388:
15384:
15378:
15375:
15370:
15366:
15361:
15358:
15352:
15346:
15338:
15337:
15318:
15312:
15305:numbers, and
15304:
15303:negafibonacci
15287:
15283:
15279:
15275:
15272:
15268:
15264:
15261:
15255:
15246:
15244:
15227:
15221:
15216:
15213:
15209:
15202:
15199:
15193:
15190:
15169:
15163:
15158:
15155:
15151:
15144:
15141:
15135:
15132:
15111:
15104:
15101:
15098:
15095:
15091:
15086:
15080:
15077:
15074:
15071:
15067:
15061:
15054:
15050:
15045:
15039:
15033:
15025:
15020:
15006:
15003:
15000:
14978:
14974:
14964:
14947:
14944:
14941:
14939:
14929:
14925:
14916:
14913:
14910:
14906:
14902:
14897:
14894:
14891:
14887:
14883:
14878:
14874:
14860:
14857:
14854:
14850:
14846:
14841:
14837:
14833:
14830:
14825:
14821:
14817:
14814:
14809:
14805:
14801:
14798:
14796:
14786:
14782:
14776:
14773:
14770:
14766:
14755:
14752:
14749:
14745:
14741:
14736:
14732:
14726:
14723:
14720:
14716:
14705:
14702:
14699:
14695:
14691:
14686:
14682:
14676:
14672:
14661:
14658:
14655:
14651:
14647:
14645:
14635:
14632:
14629:
14625:
14619:
14615:
14604:
14601:
14598:
14594:
14590:
14585:
14582:
14579:
14575:
14569:
14565:
14554:
14551:
14548:
14544:
14540:
14535:
14531:
14525:
14521:
14510:
14507:
14504:
14500:
14496:
14494:
14486:
14480:
14472:
14468:
14464:
14461:
14458:
14455:
14423:
14419:
14415:
14412:
14409:
14406:
14394:
14381:
14373:
14369:
14365:
14362:
14359:
14356:
14352:
14347:
14341:
14335:
14327:
14313:
14310:
14306:
14302:
14299:
14291:
14266:
14259:
14254:
14241:
14238:
14235:
14230:
14226:
14222:
14219:
14214:
14210:
14206:
14203:
14198:
14194:
14190:
14187:
14182:
14178:
14174:
14171:
14168:
14165:
14162:
14157:
14153:
14147:
14143:
14132:
14129:
14126:
14122:
14118:
14112:
14106:
14098:
14097:
14093:
14083:
14081:
14075:
14069:
14056:
14051:
14048:
14045:
14038:
14035:
14032:
14028:
14020:
14013:
14009:
14001:
13998:
13995:
13991:
13979:
13976:
13963:
13958:
13955:
13952:
13948:
13944:
13939:
13936:
13933:
13930:
13926:
13917:
13914:
13901:
13896:
13893:
13890:
13883:
13880:
13877:
13873:
13865:
13858:
13854:
13846:
13843:
13840:
13836:
13824:
13821:
13808:
13803:
13800:
13797:
13793:
13789:
13784:
13781:
13778:
13775:
13771:
13762:
13759:
13757:
13753:
13749:
13732:
13726:
13719:
13716:
13713:
13709:
13703:
13700:
13695:
13688:
13684:
13677:
13671:
13664:
13660:
13654:
13651:
13647:
13641:
13634:
13630:
13624:
13621:
13616:
13609:
13606:
13603:
13599:
13592:
13586:
13583:
13580:
13576:
13570:
13566:
13562:
13559:
13554:
13551:
13547:
13525:
13518:
13514:
13508:
13503:
13499:
13493:
13486:
13483:
13480:
13476:
13470:
13467:
13462:
13455:
13452:
13449:
13445:
13439:
13434:
13431:
13428:
13425:
13421:
13399:
13392:
13388:
13382:
13377:
13370:
13366:
13358:
13355:
13352:
13348:
13344:
13341:
13336:
13329:
13326:
13323:
13319:
13313:
13308:
13305:
13302:
13299:
13295:
13286:
13270:
13266:
13260:
13252:
13249:
13243:
13240:
13235:
13228:
13224:
13218:
13215:
13210:
13207:
13204:
13200:
13194:
13191:
13188:
13184:
13178:
13174:
13170:
13167:
13162:
13155:
13151:
13145:
13142:
13137:
13134:
13130:
13117:
13107:
13103:
13084:
13080:
13074:
13070:
13066:
13062:
13056:
13053:
13050:
13046:
13042:
13037:
13034:
13031:
13027:
13022:
13016:
13012:
13008:
13003:
12996:
12993:
12990:
12986:
12980:
12975:
12968:
12965:
12962:
12958:
12952:
12947:
12944:
12940:
12918:
12915:
12912:
12908:
12902:
12894:
12891:
12885:
12880:
12876:
12870:
12867:
12864:
12860:
12856:
12851:
12848:
12845:
12841:
12835:
12831:
12817:
12802:
12795:
12791:
12783:
12780:
12777:
12769:
12766:
12760:
12755:
12752:
12749:
12745:
12739:
12736:
12733:
12729:
12725:
12720:
12713:
12709:
12684:
12681:
12678:
12670:
12667:
12661:
12656:
12653:
12650:
12646:
12640:
12637:
12634:
12630:
12626:
12621:
12614:
12610:
12597:
12587:
12579:
12563:
12558:
12555:
12552:
12549:
12529:
12526:
12523:
12520:
12517:
12490:
12487:
12482:
12479:
12476:
12472:
12463:
12458:
12456:
12445:
12442:
12439:
12433:
12428:
12425:
12422:
12418:
12414:
12409:
12406:
12403:
12399:
12395:
12393:
12382:
12379:
12374:
12371:
12366:
12363:
12360:
12356:
12352:
12349:
12346:
12341:
12338:
12335:
12331:
12327:
12321:
12319:
12308:
12305:
12300:
12297:
12292:
12289:
12286:
12282:
12275:
12269:
12266:
12261:
12258:
12253:
12250:
12247:
12243:
12236:
12234:
12223:
12220:
12217:
12212:
12209:
12204:
12201:
12198:
12194:
12187:
12181:
12178:
12175:
12170:
12167:
12162:
12159:
12156:
12152:
12145:
12143:
12134:
12128:
12124:
12120:
12117:
12114:
12109:
12105:
12101:
12098:
12095:
12092:
12088:
12084:
12079:
12075:
12071:
12068:
12065:
12060:
12056:
12052:
12049:
12044:
12041:
12015:
11993:
11990:
11985:
11982:
11979:
11975:
11971:
11966:
11962:
11956:
11951:
11948:
11945:
11941:
11931:
11917:
11912:
11908:
11904:
11899:
11895:
11891:
11886:
11882:
11876:
11860:
11845:
11842:
11839:
11835:
11829:
11826:
11823:
11819:
11815:
11810:
11805:
11801:
11795:
11792:
11789:
11784:
11781:
11778:
11774:
11753:
11747:
11744:
11741:
11737:
11733:
11728:
11724:
11719:
11713:
11710:
11707:
11703:
11699:
11694:
11687:
11684:
11681:
11677:
11671:
11666:
11661:
11657:
11651:
11646:
11643:
11640:
11636:
11613:
11610:
11607:
11603:
11597:
11593:
11589:
11584:
11579:
11575:
11569:
11564:
11561:
11558:
11554:
11531:
11524:
11521:
11518:
11514:
11502:
11489:
11486:
11481:
11478:
11475:
11471:
11467:
11462:
11458:
11452:
11449:
11446:
11441:
11438:
11435:
11431:
11411:
11408:
11405:
11382:
11379:
11374:
11371:
11368:
11364:
11360:
11355:
11352:
11349:
11345:
11341:
11336:
11333:
11330:
11326:
11322:
11317:
11313:
11307:
11302:
11299:
11296:
11292:
11284:
11283:
11282:
11266:
11263:
11260:
11256:
11235:
11232:
11227:
11224:
11221:
11217:
11213:
11208:
11204:
11198:
11193:
11190:
11187:
11183:
11173:
11171:
11161:
11148:
11140:
11136:
11132:
11129:
11126:
11123:
11119:
11110:
11092:
11088:
11082:
11078:
11067:
11064:
11061:
11057:
11048:
11043:
11029:
11026:
11023:
11015:
11011:
10995:
10949:
10946:
10938:
10934:
10911:
10908:
10898:
10894:
10883:The sequence
10876:
10874:
10869:
10867:
10849:
10845:
10841:
10838:
10835:
10832:
10829:
10824:
10821:
10818:
10814:
10810:
10805:
10801:
10778:
10775:
10772:
10768:
10764:
10759:
10755:
10744:
10720:
10716:
10693:
10690:
10687:
10683:
10677:
10673:
10669:
10664:
10659:
10655:
10649:
10644:
10641:
10638:
10634:
10624:
10620:
10600:
10597:
10593:
10584:
10578:
10558:
10555:
10552:
10549:
10545:
10536:
10520:
10517:
10512:
10509:
10506:
10503:
10499:
10495:
10490:
10487:
10483:
10477:
10472:
10469:
10466:
10462:
10439:
10436:
10432:
10428:
10423:
10420:
10417:
10414:
10410:
10404:
10401:
10398:
10393:
10390:
10387:
10383:
10373:
10359:
10356:
10351:
10348:
10345:
10341:
10337:
10332:
10328:
10322:
10317:
10314:
10311:
10307:
10286:
10283:
10278:
10274:
10239:
10236:
10233:
10230:
10227:
10224:
10221:
10218:
10215:
10201:
10187:
10184:
10181:
10178:
10175:
10172:
10169:
10166:
10163:
10160:
10157:
10143:
10140:
10137:
10134:
10131:
10126:
10123:
10120:
10116:
10112:
10107:
10103:
10099:
10094:
10091:
10088:
10084:
10076:
10075:
10074:
10071:
10051:
10048:
10045:
10042:
10039:
10036:
10033:
10030:
10027:
10018:
10009:
10006:
10003:
10000:
9997:
9994:
9991:
9988:
9985:
9982:
9979:
9953:
9950:
9947:
9944:
9941:
9935:
9932:
9929:
9926:
9923:
9920:
9917:
9911:
9905:
9902:
9899:
9896:
9893:
9870:
9867:
9864:
9855:
9842:
9839:
9834:
9831:
9828:
9824:
9820:
9815:
9811:
9805:
9800:
9797:
9794:
9790:
9779:
9768:
9752:
9748:
9725:
9722:
9719:
9715:
9711:
9706:
9703:
9700:
9696:
9673:
9670:
9667:
9663:
9640:
9637:
9634:
9630:
9609:
9606:
9603:
9583:
9580:
9577:
9549:
9546:
9543:
9540:
9534:
9531:
9528:
9525:
9522:
9516:
9510:
9507:
9504:
9501:
9498:
9484:
9473:
9470:
9467:
9464:
9458:
9455:
9452:
9449:
9446:
9440:
9434:
9431:
9428:
9425:
9422:
9408:
9403:
9399:
9376:
9372:
9349:
9346:
9343:
9339:
9335:
9330:
9327:
9324:
9320:
9316:
9311:
9307:
9272:
9269:
9266:
9260:
9254:
9251:
9248:
9245:
9242:
9236:
9230:
9227:
9224:
9221:
9218:
9212:
9206:
9203:
9200:
9197:
9194:
9188:
9182:
9179:
9176:
9173:
9170:
9167:
9164:
9150:
9147:
9144:
9139:
9135:
9127:
9102:
9099:
9096:
9090:
9084:
9081:
9078:
9072:
9066:
9063:
9060:
9057:
9054:
9040:
9037:
9034:
9029:
9025:
9017:
8992:
8986:
8980:
8977:
8974:
8960:
8957:
8954:
8949:
8945:
8937:
8912:
8898:
8895:
8892:
8887:
8883:
8875:
8839:
8836:
8833:
8828:
8824:
8816:
8786:
8783:
8780:
8775:
8771:
8763:
8762:
8761:
8759:
8755:
8733:
8730:
8727:
8701:
8698:
8693:
8689:
8668:
8665:
8660:
8656:
8633:
8629:
8608:
8605:
8602:
8580:
8576:
8567:
8552:
8550:
8540:
8536:
8531:
8526:
8509:
8504:
8500:
8491:
8487:
8483:
8478:
8475:
8472:
8468:
8464:
8458:
8456:
8446:
8442:
8433:
8429:
8425:
8420:
8417:
8414:
8410:
8406:
8400:
8398:
8388:
8384:
8375:
8372:
8369:
8365:
8361:
8356:
8353:
8350:
8346:
8339:
8337:
8327:
8324:
8316:
8313:
8309:
8299:
8292:
8289:
8286:
8282:
8276:
8271:
8264:
8260:
8254:
8252:
8245:
8242:
8239:
8236:
8232:
8218:
8214:
8208:
8191:
8186:
8183:
8180:
8176:
8172:
8170:
8163:
8159:
8153:
8150:
8147:
8143:
8139:
8134:
8131:
8128:
8124:
8118:
8114:
8106:
8101:
8098:
8095:
8092:
8089:
8085:
8081:
8079:
8071:
8068:
8065:
8061:
8053:
8050:
8047:
8043:
8038:
8032:
8028:
8020:
8016:
8000:
7991:
7987:
7982:
7978:
7977:square matrix
7973:
7969:
7966:
7960:
7947:
7942:
7935:
7931:
7925:
7920:
7917:
7914:
7910:
7904:
7901:
7898:
7894:
7890:
7885:
7877:
7874:
7863:
7858:
7856:
7850:
7846:
7836:
7823:
7818:
7810:
7807:
7804:
7800:
7792:
7788:
7778:
7774:
7766:
7763:
7760:
7756:
7749:
7744:
7739:
7733:
7727:
7722:
7715:
7710:
7704:
7693:
7689:
7685:
7679:
7675:
7668:
7656:
7652:
7646:
7642:
7637:
7633:
7624:
7619:
7606:
7588:
7585:
7582:
7570:
7559:
7556:
7544:
7533:
7530:
7518:
7507:
7504:
7501:
7498:
7490:
7484:
7479:
7477:
7473:
7468:
7462:
7449:
7440:
7425:
7422:
7414:
7411:
7405:
7400:
7396:
7384:
7379:
7375:
7365:
7348:
7337:
7334:
7321:
7315:
7310:
7307:
7298:
7295:
7291:
7285:
7279:
7271:
7267:
7260:
7252:
7249:
7241:
7238:
7230:
7223:
7216:
7212:
7205:
7198:
7192:
7187:
7178:
7175:
7171:
7167:
7162:
7156:
7151:
7149:
7131:
7127:
7121:
7117:
7103:
7100:
7096:
7090:
7082:
7079:
7071:
7068:
7060:
7053:
7046:
7042:
7035:
7030:
7027:
7025:
7007:
7003:
6997:
6993:
6979:
6976:
6972:
6966:
6958:
6955:
6953:
6935:
6931:
6925:
6921:
6907:
6903:
6899:
6897:
6882:
6878:
6872:
6869:
6866:
6862:
6841:
6834:
6821:
6816:
6810:
6805:
6796:
6793:
6789:
6785:
6780:
6774:
6769:
6766:
6762:
6757:
6748:
6745:
6741:
6737:
6732:
6725:
6720:
6714:
6709:
6698:
6695:
6678:
6673:
6670:
6666:
6660:
6652:
6649:
6647:
6640:
6636:
6628:
6623:
6620:
6616:
6609:
6606:
6604:
6599:
6587:
6585:
6580:
6575:
6570:
6557:
6552:
6546:
6538:
6524:
6519:
6516:
6505:
6494:
6481:
6469:
6464:
6458:
6450:
6436:
6431:
6428:
6417:
6406:
6393:
6382:
6377:
6373:
6364:
6355:
6338:
6327:
6321:
6318:
6314:
6310:
6296:
6290:
6282:
6268:
6263:
6260:
6249:
6238:
6225:
6213:
6201:
6198:
6185:
6179:
6171:
6157:
6152:
6149:
6138:
6127:
6114:
6103:
6101:
6087:
6079:
6076:
6068:
6065:
6056:
6052:
6047:
6038:
6030:
6026:
6019:
6015:
6010:
6008:
5994:
5986:
5982:
5975:
5971:
5966:
5957:
5949:
5945:
5938:
5934:
5929:
5927:
5920:
5910:
5889:
5876:
5867:
5858:
5854:
5849:
5840:
5831:
5827:
5822:
5811:
5808:
5797:
5792:
5782:
5769:
5756:
5745:
5739:
5736:
5732:
5728:
5716:
5707:
5700:
5689:
5686:
5675:
5666:
5656:
5627:
5622:
5619:
5606:
5603:
5597:
5592:
5589:
5585:
5581:
5578:
5575:
5542:
5537:
5534:
5521:
5518:
5512:
5509:
5500:
5496:
5492:
5474:
5464:
5455:
5445:
5440:
5430:
5419:which yields
5417:
5404:
5399:
5389:
5377:
5372:
5369:
5366:
5356:
5322:
5318:
5312:
5309:
5306:
5302:
5288:
5282:
5277:
5270:
5265:
5259:
5254:
5241:
5238:
5235:
5231:
5225:
5222:
5219:
5215:
5198:
5196:
5186:
5169:
5163:
5153:
5150:
5147:
5143:
5139:
5136:
5131:
5127:
5120:
5115:
5107:
5104:
5098:
5095:
5090:
5083:
5079:
5073:
5066:
5065:
5064:
5050:
5046:
5037:
5034:
5031:
5027:
5023:
5020:
5015:
5011:
5007:
5002:
4995:
4991:
4984:
4979:
4976:
4973:
4969:
4965:
4962:
4957:
4953:
4949:
4944:
4940:
4930:
4917:
4912:
4904:
4896:
4893:
4887:
4884:
4879:
4872:
4868:
4862:
4857:
4852:
4845:
4841:
4834:
4829:
4825:
4816:
4814:
4796:
4792:
4783:
4765:
4761:
4755:
4731:
4725:
4716:
4713:
4709:
4703:
4695:
4692:
4686:
4681:
4677:
4670:
4665:
4661:
4652:
4636:
4633:
4628:
4624:
4620:
4600:
4597:
4592:
4588:
4584:
4576:
4562:
4549:
4544:
4541:
4538:
4534:
4530:
4525:
4522:
4519:
4515:
4511:
4506:
4502:
4493:
4489:
4480:
4474:
4461:
4456:
4453:
4450:
4446:
4442:
4439:
4434:
4430:
4426:
4421:
4417:
4396:
4393:
4390:
4387:
4382:
4378:
4357:
4353:
4349:
4346:
4343:
4340:
4320:
4315:
4311:
4307:
4304:
4299:
4296:
4293:
4289:
4285:
4280:
4276:
4272:
4269:
4261:
4258:
4255:
4251:
4247:
4242:
4238:
4231:
4228:
4223:
4220:
4217:
4213:
4209:
4203:
4200:
4197:
4189:
4185:
4181:
4178:
4173:
4170:
4167:
4163:
4159:
4154:
4150:
4144:
4140:
4136:
4133:
4125:
4122:
4119:
4115:
4111:
4108:
4103:
4099:
4092:
4087:
4084:
4081:
4077:
4066:
4061:
4057:
4041:
4036:
4033:
4030:
4026:
4022:
4019:
4014:
4010:
4006:
4001:
3997:
3988:
3984:
3968:
3946:
3942:
3932:
3919:
3916:
3913:
3910:
3907:
3902:
3898:
3877:
3873:
3872:
3871:
3857:
3835:
3831:
3827:
3820:
3816:
3810:
3807:
3804:
3800:
3786:
3772:
3770:
3754:
3750:
3744:
3740:
3736:
3733:
3728:
3724:
3701:
3697:
3674:
3670:
3660:
3647:
3644:
3641:
3634:
3630:
3624:
3621:
3618:
3614:
3600:
3576:
3573:
3565:
3561:
3552:
3539:
3533:
3530:
3527:
3522:
3519:
3516:
3513:
3507:
3504:
3501:
3496:
3492:
3488:
3478:
3474:
3466:
3461:
3453:
3436:
3432:
3429:
3426:
3423:
3418:
3414:
3410:
3400:
3396:
3377:
3371:
3365:
3361:
3352:
3348:
3335:
3321:
3318:
3315:
3309:
3303:
3298:
3294:
3273:
3270:
3267:
3261:
3255:
3252:
3229:
3223:
3218:
3214:
3209:
3202:
3196:
3191:
3187:
3183:
3177:
3171:
3168:
3164:
3157:
3151:
3148:
3145:
3139:
3133:
3128:
3124:
3103:
3100:
3097:
3094:
3088:
3084:
3077:
3073:
3069:
3066:
3063:
3055:
3050:
3045:
3041:
3036:
3032:
3026:
2994:
2981:
2976:
2963:
2960:
2957:
2951:
2947:
2943:
2938:
2933:
2928:
2924:
2919:
2915:
2909:
2903:
2889:
2882:
2872:
2859:
2856:
2853:
2847:
2843:
2837:
2831:
2827:
2821:
2817:
2812:
2808:
2799:
2780:
2774:
2770:
2759:
2753:
2749:
2745:, the number
2742:
2723:
2720:
2715:
2711:
2705:
2699:
2695:
2689:
2674:
2657:
2651:
2644:
2640:
2636:
2633:
2628:
2624:
2617:
2615:
2610:
2603:
2597:
2592:
2587:
2583:
2579:
2574:
2570:
2563:
2561:
2556:
2544:
2541:
2526:
2522:
2518:
2515:
2510:
2506:
2502:
2499:
2494:
2490:
2481:
2475:
2466:
2460:
2457:
2444:
2441:
2438:
2435:
2432:
2428:
2422:
2418:
2413:
2407:
2404:
2401:
2397:
2392:
2389:
2381:
2378:
2358:
2355:
2353:
2348:
2345:
2342:
2339:
2336:
2329:
2326:
2324:
2319:
2316:
2313:
2306:
2297:
2285:
2281:
2271:
2261:
2247:
2230:
2225:
2222:
2219:
2215:
2211:
2206:
2203:
2200:
2196:
2192:
2190:
2180:
2177:
2174:
2170:
2166:
2163:
2158:
2155:
2152:
2148:
2144:
2141:
2136:
2133:
2130:
2126:
2122:
2119:
2114:
2111:
2108:
2104:
2100:
2097:
2095:
2082:
2079:
2076:
2072:
2068:
2063:
2060:
2057:
2053:
2046:
2043:
2035:
2032:
2029:
2025:
2021:
2016:
2013:
2010:
2006:
1999:
1996:
1994:
1984:
1980:
1976:
1973:
1968:
1964:
1960:
1957:
1955:
1948:
1944:
1931:
1928:
1913:
1909:
1905:
1902:
1897:
1893:
1889:
1886:
1881:
1877:
1868:
1857:
1840:
1835:
1832:
1829:
1825:
1821:
1816:
1813:
1810:
1806:
1802:
1800:
1793:
1789:
1781:
1776:
1773:
1770:
1766:
1762:
1757:
1754:
1751:
1747:
1743:
1741:
1734:
1730:
1717:
1695:
1690:
1687:
1684:
1680:
1676:
1671:
1668:
1665:
1661:
1657:
1652:
1648:
1627:
1624:
1621:
1618:
1613:
1609:
1591:
1578:
1572:
1569:
1566:
1563:
1556:
1553:
1545:
1542:
1536:
1531:
1527:
1520:
1514:
1507:
1504:
1496:
1493:
1487:
1482:
1478:
1471:
1466:
1462:
1453:
1437:
1434:
1430:
1426:
1423:
1420:
1411:
1398:
1395:
1392:
1388:
1385:
1382:
1377:
1374:
1369:
1366:
1363:
1360:
1357:
1354:
1349:
1343:
1338:
1335:
1329:
1326:
1318:
1316:
1308:
1303:
1290:
1287:
1283:
1280:
1275:
1269:
1264:
1261:
1255:
1252:
1244:
1241:
1228:
1222:
1215:
1211:
1207:
1202:
1198:
1191:
1185:
1182:
1179:
1172:
1168:
1164:
1159:
1155:
1148:
1143:
1139:
1130:
1128:
1124:
1120:
1116:
1112:
1108:
1104:
1093:
1081:
1074:
1070:
1065:
1061:
1059:
1058:Ădouard Lucas
1054:
1042:
1035:
1022:
1019:
1016:
1013:
1012:
1011:
1009:
1004:
1001:unrealistic)
1000:
996:
992:
987:
986:
974:
969:
968:
962:
957:
948:
946:
941:
939:
933:
931:
927:
926:
925:Natya Shastra
921:
914:
910:
903:
899:
891:
887:
878:
870:
865:
860:
856:
847:
843:
831:
822:
813:
807:
802:
780:
777:
774:
771:
768:
765:
762:
759:
756:
753:
750:
747:
744:
741:
738:
735:
732:
729:
726:
723:
722:
715:
711:
705:
701:
695:
691:
685:
681:
675:
671:
665:
661:
655:
651:
645:
641:
635:
631:
625:
621:
615:
611:
605:
601:
595:
591:
585:
581:
575:
571:
565:
561:
555:
551:
545:
541:
535:
531:
525:
521:
520:
517:
516:
515:
512:
503:
499:
495:is valid for
480:
477:
474:
470:
466:
461:
458:
455:
451:
447:
442:
438:
417:
414:
411:
406:
402:
398:
393:
389:
368:
365:
360:
356:
346:
342:
323:
320:
317:
313:
309:
304:
301:
298:
294:
290:
285:
281:
260:
257:
254:
249:
245:
240:
237:
234:
229:
225:
217:
209:
208:circular arcs
205:
204:golden spiral
200:
191:
189:
185:
181:
180:Lucas numbers
165:
161:
156:
154:
150:
146:
142:
138:
134:
130:
126:
123:
119:
115:
114:
108:
105:
104:
98:
94:
90:
86:
78:
73:
66:
65:
64:
57:
48:
44:
40:
33:
19:
28887:
28846:
28827:
28819:
28668:Power series
28410:Lucas number
28389:
28362:Powers of 10
28342:Cubic number
28175:
28156:Pisot number
27836:Transposable
27700:Narcissistic
27607:Digital root
27527:Super-Poulet
27487:JordanâPĂłlya
27436:prime factor
27341:Noncototient
27308:Almost prime
27290:Superperfect
27265:Refactorable
27260:Quasiperfect
27235:Hyperperfect
27076:Pseudoprimes
27047:WallâSunâSun
26982:Ordered Bell
26952:FussâCatalan
26864:non-centered
26814:Dodecahedral
26791:non-centered
26677:non-centered
26579:Wolstenholme
26442:
26324:× 2 ± 1
26321:
26320:Of the form
26287:Eighth power
26267:Fourth power
26158:
26143:
26130:at MathPages
26092:
26081:
26057:
26053:Livio, Mario
26034:
26015:
25988:
25985:BĂłna, MiklĂłs
25966:
25943:
25921:
25909:
25893:11250/180781
25875:
25871:
25865:
25831:
25825:
25816:
25809:
25797:
25787:, retrieved
25780:the original
25775:
25750:
25741:
25737:
25725:
25702:
25684:
25672:
25656:
25652:
25646:
25624:
25618:
25609:
25603:, retrieved
25588:
25581:
25569:
25557:
25540:
25533:
25524:
25515:
25504:
25498:
25480:
25474:
25463:the original
25450:
25446:
25433:
25423:, retrieved
25417:
25410:
25401:
25394:
25385:
25380:
25371:
25365:
25357:
25348:
25342:
25336:
25318:
25309:
25277:
25270:
25257:(4): 39â40,
25254:
25250:
25244:
25235:
25232:Modern Logic
25231:
25225:
25217:
25199:
25193:
25188:, p. 7.
25181:
25165:
25161:
25143:
25131:
25118:
25094:
25090:
25084:
25074:
25066:
25062:
25054:
25048:
25038:
25031:
25027:
25019:
25013:
25001:
24989:
24977:
24959:
24950:
24938:
24925:
24919:, UK: Surrey
24915:
24913:Knott, Ron,
24908:
24884:
24880:
24870:
24854:
24850:
24840:
24800:
24796:
24790:
24781:
24777:
24764:
24732:math/0403046
24722:
24718:
24712:
24703:
24699:
24693:
24669:
24665:
24659:
24641:
24635:
24625:, retrieved
24618:the original
24595:
24575:
24563:
24550:
24542:
24538:
24501:
24495:
24448:
24438:, retrieved
24434:the original
24428:
24421:
24412:
24403:
24385:
24375:
24359:
24353:
24347:
24335:
24318:
24302:
24270:
24266:
24260:
24249:
24211:
24204:
24186:
24180:
24175:, p. 4.
24168:
24156:
24146:
24135:. Retrieved
24131:
24122:
24111:, retrieved
24106:
24100:
24087:
24078:
24073:
24055:
24049:
24037:
24024:
24012:
23999:
23987:
23975:
23949:
23943:
23920:
23913:
23904:
23890:
23881:
23867:
23865:Knott, Ron,
23860:
23849:, retrieved
23843:
23833:
23815:
23809:
23798:, retrieved
23791:
23782:
23770:
23759:, retrieved
23752:
23739:
23727:
23718:
23712:
23690:
23684:
23678:
23669:
23665:
23662:
23656:
23634:ed) ...
23632:
23617:
23607:
23595:
23583:
23578:, p. 3.
23571:
23559:
23547:
23544:
23529:
23498:
23494:
23466:
23442:
23402:
23275:
23264:
23054:Y chromosome
23050:X chromosome
23047:
23030:
23026:
23018:
23014:
23007:
23003:
22999:
22976:
22961:
22957:
22953:
22942:
22938:
22934:
22930:
22926:golden angle
22907:
22836:
22825:Helmut Vogel
22814:
22807:
22786:
22778:parastichies
22758:golden ratio
22731:
22612:
22357:
22313:
22305:
22301:
22273:
22269:
22262:
22243:
22235:
22231:
22213:
22199:
22195:
22191:
22179:
22166:
22162:
22144:
22136:
22132:
22106:
22098:
22029:
22022:
21984:
21980:
21976:compositions
21969:
21966:compositions
21950:
21946:
21848:
21782:
21417:
21397:Applications
21389:
21380:
21376:
21372:
21368:
21364:
21360:
21333:
21329:
21323:
21319:
21312:
21308:Pell numbers
21293:
21289:
21283:
21279:
21272:
21262:
21252:
21241:real numbers
21227:
21212:
21189:finite field
21177:open problem
21163:
21151:
21140:
21130:
21123:
21119:
21116:
20889:
20882:
20878:
20871:
20867:
20860:
20490:
20483:
20479:
20478:
20102:
20095:
20091:
20090:
19720:
19713:
19709:
19708:
19326:
19319:
19315:
19314:
19021:
19018:
19011:
18930:
18923:
18451:
18318:
18104:
18095:
18086:
18073:
18064:
18055:
18048:
18045:
18029:
18025:Vern Hoggatt
18014:
18003:
17993:
17990:
17972:
17965:
17961:
17954:
17950:
17948:
17945:
17931:
17929:
17904:
17744:
17737:
17732:
17728:
17715:
17626:
17552:
17362:
17354:
17344:
17336:
17332:
17316:
17312:
17302: 5. If
17290:prime number
17287:
17280:
17126:. That is,
17055:
16926:
16887:
16661:
16660:
16649:
16485:
16129:
15987:golden ratio
15982:
15980:
15888:
15674:
15525:
15402:
15339:
15247:
15026:is given by
15021:
14965:
14395:
14328:
14255:
14099:
14096:power series
14089:
14073:
14070:
13918:
13915:
13763:
13760:
13287:
13116:Lucas number
13105:
13101:
12823:
12599:
12585:
11932:
11866:
11503:
11397:
11174:
11167:
11044:
10882:
10870:
10742:
10625:
10618:
10585:index up to
10576:
10537:index up to
10374:
10265:
10072:
9856:
9777:
9769:
9298:
8563:
8538:
8534:
8527:
8216:
8212:
8209:
7998:
7980:
7971:
7967:
7964:
7961:
7859:
7848:
7844:
7838:For a given
7837:
7694:
7687:
7683:
7677:
7673:
7666:
7654:
7650:
7644:
7640:
7635:
7631:
7620:
7491:
7480:
7466:
7463:
7366:
6842:
6835:
6699:
6696:
6588:
6578:
6571:
6356:
5890:
5770:
5655:eigenvectors
5498:
5418:
5199:
5192:
5184:
4931:
4817:
4568:
4484:
4478:
4475:
4064:
3987:coefficients
3933:
3889:
3774:In general,
3773:
3661:
3558:
3476:
3472:
3462:
3451:
3398:
3394:
3343:
3341:
2977:
2887:
2880:
2873:
2751:
2747:
2740:
2680:
2545:
2542:
2482:
2473:
2464:
2461:
2458:
2382:
2379:
2298:
2283:
2279:
2269:
2259:
2248:
1932:
1929:
1869:
1858:
1718:
1592:
1454:
1412:
1319:
1307:golden ratio
1304:
1245:
1242:
1131:
1114:
1100:
1092:Golden ratio
1076:
1072:
1055:
1040:
1033:
1026:
1008:math problem
999:biologically
990:
980:
943:
938:mÄtrÄ-váčttas
937:
935:
923:
920:Bharata Muni
912:
908:
901:
897:
889:
885:
876:
866:
858:
854:
839:
829:
820:
811:
713:
703:
693:
683:
673:
663:
653:
643:
633:
623:
613:
603:
593:
583:
573:
563:
553:
543:
533:
523:
507:
504:
497:
347:
340:
213:
160:golden ratio
157:
111:
109:
82:
52:
46:
38:
36:
28821:Liber Abaci
28535:Conditional
28523:Convergence
28514:Telescoping
28499:Alternating
28415:Pell number
28225:Pell number
27857:Extravagant
27852:Equidigital
27807:permutation
27766:Palindromic
27740:Automorphic
27638:Sum-product
27617:Sum-product
27572:Persistence
27467:ErdĆsâWoods
27389:Untouchable
27270:Semiperfect
27220:Hemiperfect
26881:Tesseractic
26819:Icosahedral
26799:Tetrahedral
26730:Dodecagonal
26431:Recursively
26302:Prime power
26277:Sixth power
26272:Fifth power
26252:Power of 10
26210:Classes of
26144:In Our Time
25936:Works cited
25509:(7): 525â32
25374:(3): 265â69
25238:(3): 345â55
25168:: 169â177,
25073:10000 <
25058:, Red golpe
25023:, Mersennus
24784:(2): 98â108
24672:: 537â540,
24109:(4): 417â19
23851:27 November
23800:28 November
23775:Sigler 2002
23564:Sigler 2002
22981:(which are
22828: [
22810:= 1 ... 500
22792:free groups
22782:phyllotaxis
22622:binary tree
22358:one or more
22341:his solving
22102:recursively
22051:= 1+1+1+1+1
21797:= 1+1+1+1+1
21402:Mathematics
20493:⥠1 (mod 4)
20105:⥠1 (mod 4)
19723:⥠3 (mod 4)
19329:⥠3 (mod 4)
18021:conjectured
14279:satisfying
14080:Matrix form
8754:cardinality
8549:memoization
8530:recursively
7623:convergents
7472:determinant
7464:The matrix
5895:th term is
5491:eigenvalues
5189:Matrix form
1101:Like every
993:, 1202) by
985:Liber Abaci
967:Liber Abaci
945:Hemachandra
103:Liber Abaci
28907:Categories
28560:Convergent
28504:Convergent
28069:Graphemics
27942:Pernicious
27796:Undulating
27771:Pandigital
27745:Trimorphic
27346:Nontotient
27195:Arithmetic
26809:Octahedral
26710:Heptagonal
26700:Pentagonal
26685:Triangular
26526:SierpiĆski
26448:Jacobsthal
26247:Power of 3
26242:Power of 2
25926:Livio 2003
25914:Livio 2003
25802:Livio 2003
25789:2016-09-03
25677:Livio 2003
25605:2022-10-30
25574:Livio 2003
25562:Livio 2003
25425:2022-08-15
25186:Lucas 1891
25077:< 50000
25041:< 10000
24887:: 107â24,
24719:Ann. Math.
24644:(9): 133,
24627:2022-11-23
24440:2007-02-23
24173:Lucas 1891
24137:2024-03-22
23732:Livio 2003
23649:Livio 2003
23576:Lucas 1891
23389:References
23332:Mario Merz
23295:conversion
23293:Since the
22762:pentagonal
22760:-related)
22715:See also:
22600:merge sort
22549:ring lemma
22373:hypotenuse
22203:for every
22188:for which
21944:twos from
21133:< 50000
20480:Example 4.
20092:Example 3.
19710:Example 2.
19316:Example 1.
18928:such that
18017:triangular
17281:for every
17053:function.
16652:irrational
15528:reciprocal
11014:partitions
7986:identities
7682:= 1 + 1 /
3351:asymptotic
959:A page of
877:misrau cha
809:Thirteen (
799:See also:
194:Definition
28806:Fibonacci
28591:Divergent
28509:Divergent
28371:Advanced
28347:Factorial
28295:Sequences
28186:Rectangle
27826:Parasitic
27675:Factorion
27602:Digit sum
27594:Digit sum
27412:Fortunate
27399:Primorial
27313:Semiprime
27250:Practical
27215:Descartes
27210:Deficient
27200:Betrothed
27042:Wieferich
26871:Pentatope
26834:pyramidal
26725:Decagonal
26720:Nonagonal
26715:Octagonal
26705:Hexagonal
26564:Practical
26511:Congruent
26443:Fibonacci
26407:Loeschian
26165:EMS Press
26055:(2003) ,
25351:: 263â266
25302:pp. 73â74
24893:1787-6117
24833:121789033
24817:1973-4409
24472:ε
24469:−
24295:116536130
24251:MathWorld
24113:April 11,
23992:Ball 2003
23980:Ball 2003
23761:4 January
23600:BĂłna 2011
23418:Citations
23326:economics
23302:2 number
23269:, is the
22991:drone bee
22861:φ
22856:π
22844:θ
22821:sunflower
22750:honeybees
22746:pine cone
22742:artichoke
22738:pineapple
22525:triangle.
22261:{1, ...,
22209:bijection
22178:{1, ...,
22080:= 1+1+1+2
22066:= 1+1+2+1
22061:= 1+2+1+1
22056:= 2+1+1+1
22005:×
21849:which is
21821:= 1+1+1+2
21816:= 1+1+2+1
21811:= 1+2+1+1
21806:= 2+1+1+1
21719:∞
21704:∑
21697:⋯
21650:⋯
21562:−
21556:−
21509:−
21503:−
21473:−
21448:∑
21100:⋅
21094:⋅
21012:⋅
20828:≡
20769:≡
20588:−
20446:≡
20387:−
20384:≡
20214:−
20200:−
20144:−
20058:≡
19999:≡
19885:−
19676:−
19673:≡
19614:−
19611:≡
19508:−
19494:−
19438:−
19368:−
19272:≡
19235:∓
19163:≡
19126:±
19069:≡
19049:±
19019:Also, if
18969:≡
18946:−
18757:−
18593:−
18497:−
18416:≡
18393:−
18341:≡
18276:±
18273:≡
18257:−
18231:±
18228:≡
17828:≡
17812:−
17708:composite
17654:−
17642:∣
17584:−
17572:∣
17555:piecewise
17513:∣
17507:⇒
17483:±
17480:≡
17462:−
17451:∣
17445:⇒
17421:±
17418:≡
17395:∣
17389:⇒
17322:, and if
17027:…
16989:…
16837:−
16817:−
16767:∑
16726:−
16688:∞
16673:∑
16636:…
16605:∞
16590:∑
16576:−
16555:∞
16540:∑
16514:∞
16499:∑
16450:ψ
16446:−
16430:ψ
16424:−
16409:ψ
16405:−
16389:ψ
16313:−
16292:∞
16277:∑
16249:with the
16223:ψ
16213:−
16201:ψ
16154:∞
16139:∑
16104:−
16049:∑
16024:−
16013:∞
15998:∑
15941:−
15913:∞
15898:∑
15837:−
15809:ϑ
15805:−
15778:−
15750:ϑ
15699:∞
15684:∑
15634:−
15606:ϑ
15579:−
15558:∞
15543:∑
15504:…
15371:−
15273:−
15262:−
15259:↦
15243:conjugate
15217:−
15191:ψ
15133:φ
15102:ψ
15099:−
15087:−
15078:φ
15075:−
15004:≥
14914:−
14903:−
14895:−
14884:−
14866:∞
14851:∑
14831:−
14774:−
14761:∞
14746:∑
14742:−
14724:−
14711:∞
14696:∑
14692:−
14667:∞
14652:∑
14610:∞
14595:∑
14591:−
14560:∞
14545:∑
14541:−
14516:∞
14501:∑
14465:−
14459:−
14416:−
14410:−
14366:−
14360:−
14311:φ
14239:…
14138:∞
14123:∑
14049:−
14036:−
13949:∑
13894:−
13844:−
13794:∑
13756:factorize
13652:−
13383:−
13250:−
13208:−
13054:−
12994:−
12981:−
12916:−
12892:−
12857:−
12781:−
12767:−
12753:−
12726:−
12682:−
12668:−
12654:−
12627:−
12556:ψ
12553:−
12550:φ
12527:−
12521:ψ
12518:φ
12488:−
12446:ψ
12443:−
12440:φ
12434:−
12419:ψ
12415:−
12400:φ
12383:ψ
12380:φ
12375:ψ
12372:−
12357:ψ
12350:φ
12332:φ
12328:−
12309:φ
12306:−
12298:−
12283:ψ
12276:−
12270:ψ
12267:−
12259:−
12244:φ
12221:−
12218:ψ
12210:−
12195:ψ
12188:−
12179:−
12176:φ
12168:−
12153:φ
12125:ψ
12118:⋯
12106:ψ
12099:ψ
12085:−
12076:φ
12069:⋯
12057:φ
12050:φ
11991:−
11942:∑
11909:ψ
11905:−
11896:φ
11775:∑
11637:∑
11555:∑
11487:−
11432:∑
11380:−
11293:∑
11233:−
11184:∑
11133:−
11127:−
11107:, is the
11073:∞
11058:∑
11027:−
11012:(ordered
10950:
10912:∈
10822:−
10765:×
10635:∑
10556:−
10518:−
10463:∑
10402:−
10384:∑
10357:−
10308:∑
10124:−
9840:−
9791:∑
9723:−
9704:−
9671:−
9638:−
9607:−
9581:−
9347:−
9328:−
8606:−
8484:−
8418:−
8354:−
8325:−
8290:−
8243:−
8151:−
8099:−
8069:−
8051:−
7926:−
7918:−
7875:−
7808:−
7589:⋱
7499:φ
7423:−
7415:φ
7412:−
7406:−
7397:φ
7316:φ
7308:−
7296:−
7292:φ
7250:−
7242:φ
7239:−
7213:φ
7176:−
7172:φ
7168:−
7163:φ
7101:−
7080:−
7072:φ
7069:−
7043:φ
6977:−
6963:Λ
6794:−
6790:φ
6786:−
6781:φ
6746:−
6742:φ
6738:−
6721:φ
6707:Λ
6671:−
6657:Λ
6621:−
6613:Λ
6520:−
6470:−
6319:−
6315:φ
6311:−
6264:−
6214:−
6199:φ
6091:→
6088:ν
6077:−
6069:φ
6066:−
6048:−
6042:→
6039:μ
6027:φ
5998:→
5995:ν
5967:−
5961:→
5958:μ
5914:→
5871:→
5868:ν
5850:−
5844:→
5841:μ
5786:→
5737:−
5733:φ
5729:−
5711:→
5708:ν
5687:φ
5670:→
5667:μ
5623:−
5590:−
5586:φ
5582:−
5576:ψ
5510:φ
5468:→
5434:→
5393:→
5360:→
5151:−
5105:−
5035:−
4977:−
4963:φ
4941:φ
4894:−
4858:±
4826:φ
4793:φ
4762:φ
4714:−
4710:φ
4693:−
4687:−
4678:φ
4634:−
4531:−
4454:−
4440:ψ
4418:ψ
4391:ψ
4379:ψ
4358:φ
4347:−
4341:ψ
4305:φ
4270:φ
4259:−
4229:φ
4221:−
4198:φ
4179:φ
4171:−
4151:φ
4134:φ
4123:−
4109:φ
4078:φ
4060:induction
4034:−
4020:φ
3998:φ
3969:φ
3943:φ
3911:φ
3899:φ
3858:φ
3832:φ
3793:∞
3790:→
3755:φ
3737:−
3716:, unless
3645:φ
3607:∞
3604:→
3577::
3574:φ
3564:converges
3531:
3523:φ
3520:
3505:φ
3502:
3430:≈
3427:φ
3424:
3362:φ
3338:Magnitude
3322:…
3310:φ
3304:
3274:…
3262:φ
3256:
3230:φ
3224:
3197:
3178:φ
3172:
3152:
3134:
3129:φ
3098:≥
3051:
3046:φ
2961:≥
2934:
2929:φ
2857:≥
2828:φ
2771:φ
2696:ψ
2637:−
2634:φ
2593:ψ
2580:−
2523:ψ
2507:φ
2439:−
2408:ψ
2405:−
2402:φ
2346:ψ
2337:φ
2223:−
2204:−
2178:−
2171:ψ
2156:−
2149:φ
2134:−
2127:ψ
2112:−
2105:φ
2080:−
2073:ψ
2061:−
2054:ψ
2033:−
2026:φ
2014:−
2007:φ
1981:ψ
1965:φ
1910:ψ
1894:φ
1833:−
1826:ψ
1814:−
1807:ψ
1790:ψ
1774:−
1767:φ
1755:−
1748:φ
1731:φ
1688:−
1669:−
1640:and thus
1570:−
1567:φ
1554:−
1546:φ
1543:−
1537:−
1528:φ
1505:−
1497:φ
1494:−
1488:−
1479:φ
1435:−
1431:φ
1427:−
1421:ψ
1396:…
1386:−
1383:≈
1378:φ
1370:−
1364:φ
1361:−
1339:−
1327:ψ
1315:conjugate
1291:…
1281:≈
1253:φ
1212:ψ
1208:−
1199:φ
1186:ψ
1183:−
1180:φ
1169:ψ
1165:−
1156:φ
995:Fibonacci
971:from the
961:Fibonacci
930:Virahanka
852:units is
478:−
459:−
321:−
302:−
153:pine cone
149:artichoke
145:pineapple
97:Fibonacci
28840:Theories
28764:Category
28530:Absolute
28206:Triangle
27898:Friedman
27831:Primeval
27776:Repdigit
27733:-related
27680:Kaprekar
27654:Meertens
27577:Additive
27564:dynamics
27472:Friendly
27384:Sociable
27374:Amicable
27185:Abundant
27165:dynamics
26987:Schröder
26977:Narayana
26947:Eulerian
26937:Delannoy
26932:Dedekind
26753:centered
26619:centered
26506:Amenable
26463:Narayana
26453:Leonardo
26349:Mersenne
26297:Powerful
26237:Achilles
25987:(2011),
25858:35413237
25696:(1990),
25317:(1997),
25263:30215477
24958:(1996),
24851:Integers
24757:10266596
24583:(2018),
24411:(2000),
24154:(1978),
23889:(1996),
23615:(1968),
23527:(2006),
23353:See also
23304:register
22960:(
22941:(
22933:(
22780:(spiral
22673:compands
22630:AVL tree
21485:⌋
21464:⌊
21236:numbers.
21158:periodic
21128:for all
20861:For odd
19265:if
19156:if
18266:if
18221:if
18198:if
17984:runs of
17718:. When
17353:divides
17330:divides
17310:divides
15480:0.998999
14071:Putting
12030:becomes
11628:to give
10739:-th and
7975:for any
7857:method.
4811:via the
3319:0.208987
3271:0.481211
3085:⌋
3037:⌊
2948:⌉
2920:⌊
2844:⌉
2822:⌊
2798:rounding
2738:for all
1103:sequence
120:and the
93:Sanskrit
43:sequence
28861:Related
28540:Uniform
28191:Rhombus
28071:related
28035:related
27999:related
27997:Sorting
27882:Vampire
27867:Harshad
27809:related
27781:Repunit
27695:Lychrel
27670:Dudeney
27522:StĂžrmer
27517:Sphenic
27502:Regular
27440:divisor
27379:Perfect
27275:Sublime
27245:Perfect
26972:Motzkin
26927:Catalan
26468:Padovan
26402:Leyland
26397:Idoneal
26392:Hilbert
26364:Woodall
26167:, 2001
26147:at the
26122:YouTube
25902:8550726
25838:Bibcode
25708:101â107
25399:"IFF",
25297:2131318
25174:2278830
25126:(ed.),
25111:2325076
24933:(ed.),
24901:2753031
24863:2988067
24825:1765401
24737:Bibcode
24706:: 81â96
24686:0163867
24614:3821829
24558:(ed.),
24530:0668957
24368:0999451
24326:(ed.),
24311:Borwein
24287:3618079
24032:(ed.),
24007:(ed.),
23697:: 232,
23437:(ed.),
22817:florets
22766:daisies
22660:in the
22198:+ 1} â
22116:subsets
22112:strings
22090:= 1+2+2
22085:= 2+1+2
22071:= 2+2+1
21994:of the
21955:terms.
21840:= 1+2+2
21835:= 2+1+2
21830:= 2+2+1
21375:â 2) +
20825:1860500
17058:coprime
17049:is the
16486:So the
15241:is its
13110:is the
11248:Adding
10615:is the
10573:is the
8752:is the
7659:is the
6697:where
5493:of the
5489:. The
2758:integer
1389:0.61803
1313:is its
1305:is the
1284:1.61803
1069:problem
918:cases.
883:beats (
869:Pingala
790:History
131:called
89:Pingala
77:squares
60:
28832:(1225)
28824:(1202)
28492:Series
28299:series
28245:etc...
28242:Nickel
28239:Copper
28236:Bronze
28220:Silver
28201:Spiral
27937:Odious
27862:Frugal
27816:Cyclic
27805:Digit-
27512:Smooth
27497:Pronic
27457:Cyclic
27434:Other
27407:Euclid
27057:Wilson
27031:Primes
26690:Square
26559:Polite
26521:Riesel
26516:Knödel
26478:Perrin
26359:Thabit
26344:Fermat
26334:Cullen
26257:Square
26225:Powers
26099:
26069:
26041:
26023:
25995:
25973:
25954:
25900:
25856:
25714:
25635:
25596:
25487:
25325:
25295:
25285:
25261:
25206:
25172:
25109:
24966:
24899:
24891:
24861:
24857:: A7,
24831:
24823:
24815:
24755:
24684:
24648:
24612:
24602:
24528:
24392:
24366:
24307:Landau
24293:
24285:
24219:
24193:
24081:, 1611
24062:
23964:
23928:
23897:
23822:
23625:
23537:
23474:
23321:ratio.
23267:> 1
23252:optics
22908:where
22878:
22754:Kepler
22705:Nature
22626:height
22532:is an
22109:binary
21270:, and
21065:
21043:
21021:
20983:
20961:
20939:
20917:
20766:710645
18006:square
17300:modulo
17288:Every
17045:where
16342:since
15983:nested
15493:998999
15403:Using
15125:where
13099:where
7470:has a
5633:
5548:
5495:matrix
4481:< 1
4056:proved
3454:> 1
3433:0.2090
3342:Since
3286:, and
3116:where
3092:
2955:
2851:
2681:Since
2543:where
1413:Since
1309:, and
1243:where
1003:rabbit
951:Europe
500:> 2
343:> 1
129:graphs
127:, and
28813:Books
28435:array
28315:Basic
28166:Angle
27978:Prime
27973:Lucky
27962:sieve
27891:Other
27877:Smith
27757:Digit
27715:Happy
27690:Keith
27663:Other
27507:Rough
27477:Giuga
26942:Euler
26804:Cubic
26458:Lucas
26354:Proth
25898:S2CID
25854:S2CID
25783:(PDF)
25772:(PDF)
25734:(PDF)
25545:(PDF)
25525:Maths
25466:(PDF)
25443:(PDF)
25418:Story
25259:JSTOR
25107:JSTOR
25071:with
25036:with
24829:S2CID
24774:(PDF)
24753:S2CID
24727:arXiv
24621:(PDF)
24590:, in
24588:(PDF)
24291:S2CID
24283:JSTOR
24161:(PDF)
24097:(PDF)
23906:abaci
23749:(PDF)
23693:(3),
23408:morae
23300:radix
23245:Other
22922:angle
22832:]
22677:Ό-law
22669:Amiga
22375:of a
22325:s or
22298:s is
22294:s or
21358:have
21310:have
21250:have
21183:of a
17936:prime
17736:(mod
17714:is a
15477:0.001
15466:0.001
11016:) of
10866:areas
8756:of a
8537:(log
7996:into
7847:(log
4649:is a
1393:39887
1288:39887
795:India
781:4181
778:2584
775:1597
514:are:
41:is a
28375:list
28297:and
28171:Base
28161:Gold
27932:Evil
27612:Self
27562:and
27452:Blum
27163:and
26967:Lobb
26922:Cake
26917:Bell
26667:Star
26574:Ulam
26473:Pell
26262:Cube
26172:OEIS
26097:ISBN
26067:ISBN
26039:ISBN
26021:ISBN
25993:ISBN
25971:ISBN
25952:ISBN
25712:ISBN
25633:ISBN
25594:ISBN
25485:ISBN
25323:ISBN
25283:ISBN
25204:ISBN
25132:The
24964:ISBN
24939:The
24889:ISSN
24813:ISSN
24646:ISBN
24600:ISBN
24564:The
24390:ISBN
24336:The
24217:ISBN
24191:ISBN
24115:2012
24060:ISBN
24038:The
24013:The
23962:ISBN
23926:ISBN
23895:ISBN
23853:2018
23820:ISBN
23802:2018
23763:2019
23757:: 41
23623:ISBN
23535:ISBN
23472:ISBN
23443:The
22979:bees
22945:+ 1)
22772:and
22665:8SVX
22555:and
22540:for
22528:The
22207:. A
21383:â 3)
21367:) =
21354:and
20722:610.
20486:= 29
20098:= 13
19716:= 11
18093:and
18071:and
17093:and
15490:1000
15022:The
14993:for
14300:<
14090:The
13114:-th
12542:and
10745:+ 1)
10621:+ 1)
10583:even
10454:and
9780:+ 2)
7621:The
5568:and
5502:are
4333:For
3689:and
3465:base
2716:<
2471:and
2292:and
2267:and
2253:and
1863:and
1712:and
1597:and
1125:and
772:987
769:610
766:377
763:233
760:144
338:for
273:and
28050:Ban
27438:or
26957:Lah
26149:BBC
26141:on
26120:on
25888:hdl
25880:doi
25876:154
25846:doi
25698:"4"
25661:doi
25455:doi
25451:178
25349:146
25099:doi
24855:11a
24805:doi
24745:doi
24674:doi
24516:hdl
24506:doi
24494:",
24360:308
24275:doi
23954:doi
23699:doi
23550:= 7
23503:doi
23306:in
23278:= 1
23250:In
22662:IFF
22587:of
22319:= 6
22259:of
22249:= 5
22176:of
22150:= 8
22035:= 8
22026:5+1
21318:= 2
21268:= 3
21258:= 1
21103:61.
21083:610
21059:233
20839:mod
20780:mod
20701:377
20457:mod
20443:845
20398:mod
20381:320
20337:13.
20069:mod
20055:320
20010:mod
19996:125
19687:mod
19625:mod
19322:= 7
19283:mod
19251:mod
19174:mod
19142:mod
19024:â 5
18980:mod
18906:89.
18427:mod
18380:and
18367:mod
18319:If
18287:mod
18242:mod
18023:by
17999:= 8
17978:= 3
17879:mod
17724:bit
17706:is
17494:mod
17432:mod
17347:= 5
17219:gcd
17178:gcd
17137:gcd
17047:gcd
17003:gcd
16944:gcd
16654:by
16490:is
14076:= 2
13754:to
10947:Seq
10535:odd
9655:or
9596:or
8758:set
8551:).
8532:in
8003:),
8001:+ 1
6576:of
4784:in
4613:or
4490:is
4067:â„ 1
4062:on
4058:by
3783:lim
3597:lim
3528:log
3517:log
3493:log
3415:log
3353:to
3349:is
3295:log
3215:log
3188:log
3125:log
3042:log
2925:log
2890:â„ 8
2883:â„ 4
2760:to
2743:â„ 0
2275:= 1
2265:= 0
2249:If
1043:â 1
1036:â 2
963:'s
757:89
754:55
751:34
748:21
745:13
502:.
28909::
26163:,
26157:,
26065:,
26009:;
25950:,
25896:,
25886:,
25874:,
25852:,
25844:,
25774:,
25759:^
25740:,
25736:,
25710:,
25700:,
25692:;
25657:44
25655:,
25631:,
25627:,
25608:,
25547:,
25523:,
25449:,
25445:,
25293:MR
25291:,
25255:30
25253:,
25234:,
25230:,
25170:MR
25166:71
25164:,
25160:,
25130:,
25105:,
25095:99
25093:,
24937:,
24897:MR
24895:,
24885:37
24883:,
24879:,
24859:MR
24853:,
24849:,
24827:,
24821:MR
24819:,
24811:,
24801:49
24799:,
24782:27
24780:,
24776:,
24751:,
24743:,
24735:,
24721:,
24704:17
24702:,
24682:MR
24680:,
24670:39
24668:,
24610:MR
24608:,
24562:,
24526:MR
24524:,
24514:,
24502:25
24500:,
24384:,
24364:MR
24358:,
24334:,
24289:,
24281:,
24271:79
24269:,
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6579:A
6558:.
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5946:A
5939:5
5935:1
5930:=
5921:n
5911:F
5893:n
5877:,
5859:5
5855:1
5832:5
5828:1
5823:=
5817:)
5812:0
5809:1
5804:(
5798:=
5793:0
5783:F
5757:.
5751:)
5746:1
5740:1
5723:(
5717:=
5701:,
5695:)
5690:1
5682:(
5676:=
5639:)
5628:5
5620:1
5615:(
5607:2
5604:1
5598:=
5593:1
5579:=
5554:)
5543:5
5538:+
5535:1
5530:(
5522:2
5519:1
5513:=
5499:A
5475:0
5465:F
5456:n
5451:A
5446:=
5441:n
5431:F
5405:,
5400:k
5390:F
5382:A
5378:=
5373:1
5370:+
5367:k
5357:F
5330:)
5323:k
5319:F
5313:1
5310:+
5307:k
5303:F
5297:(
5289:)
5283:0
5278:1
5271:1
5266:1
5260:(
5255:=
5249:)
5242:1
5239:+
5236:k
5232:F
5226:2
5223:+
5220:k
5216:F
5210:(
5170:.
5164:2
5160:)
5154:1
5148:n
5144:F
5140:2
5137:+
5132:n
5128:F
5124:(
5121:=
5116:n
5112:)
5108:1
5102:(
5099:4
5096:+
5091:2
5084:n
5080:F
5074:5
5051:2
5047:/
5043:)
5038:1
5032:n
5028:F
5024:2
5021:+
5016:n
5012:F
5008:+
5003:5
4996:n
4992:F
4988:(
4985:=
4980:1
4974:n
4970:F
4966:+
4958:n
4954:F
4950:=
4945:n
4918:.
4913:2
4905:n
4901:)
4897:1
4891:(
4888:4
4885:+
4880:2
4873:n
4869:F
4863:5
4853:5
4846:n
4842:F
4835:=
4830:n
4797:n
4766:n
4756:5
4732:5
4726:/
4722:)
4717:n
4704:n
4700:)
4696:1
4690:(
4682:n
4674:(
4671:=
4666:n
4662:F
4637:4
4629:2
4625:x
4621:5
4601:4
4598:+
4593:2
4589:x
4585:5
4571:x
4550:.
4545:1
4542:+
4539:n
4535:F
4526:2
4523:+
4520:n
4516:F
4512:=
4507:n
4503:F
4487:n
4485:F
4479:n
4462:.
4457:1
4451:n
4447:F
4443:+
4435:n
4431:F
4427:=
4422:n
4397:1
4394:+
4388:=
4383:2
4354:/
4350:1
4344:=
4321:.
4316:n
4312:F
4308:+
4300:1
4297:+
4294:n
4290:F
4286:=
4281:n
4277:F
4273:+
4267:)
4262:1
4256:n
4252:F
4248:+
4243:n
4239:F
4235:(
4232:=
4224:1
4218:n
4214:F
4210:+
4207:)
4204:1
4201:+
4195:(
4190:n
4186:F
4182:=
4174:1
4168:n
4164:F
4160:+
4155:2
4145:n
4141:F
4137:=
4131:)
4126:1
4120:n
4116:F
4112:+
4104:n
4100:F
4096:(
4093:=
4088:1
4085:+
4082:n
4065:n
4042:.
4037:1
4031:n
4027:F
4023:+
4015:n
4011:F
4007:=
4002:n
3947:n
3920:,
3917:1
3914:+
3908:=
3903:2
3836:m
3828:=
3821:n
3817:F
3811:m
3808:+
3805:n
3801:F
3787:n
3751:/
3745:0
3741:U
3734:=
3729:1
3725:U
3702:1
3698:U
3675:0
3671:U
3648:.
3642:=
3635:n
3631:F
3625:1
3622:+
3619:n
3615:F
3601:n
3540:.
3534:b
3514:n
3508:=
3497:b
3489:n
3477:n
3473:F
3468:b
3458:d
3452:d
3437:n
3411:n
3399:n
3395:F
3378:5
3372:/
3366:n
3346:n
3344:F
3316:=
3313:)
3307:(
3268:=
3265:)
3259:(
3233:)
3227:(
3210:/
3206:)
3203:x
3200:(
3184:=
3181:)
3175:(
3165:/
3161:)
3158:x
3155:(
3146:=
3143:)
3140:x
3137:(
3104:,
3101:0
3095:F
3089:,
3081:)
3078:2
3074:/
3070:1
3067:+
3064:F
3061:(
3056:5
3033:=
3030:)
3027:F
3024:(
3018:t
3015:s
3012:e
3009:g
3006:r
3003:a
3000:l
2995:n
2984:F
2958:F
2952:,
2944:F
2939:5
2916:=
2913:)
2910:F
2907:(
2904:n
2894:F
2888:n
2881:n
2876:n
2854:n
2848:,
2838:5
2832:n
2818:=
2813:n
2809:F
2781:5
2775:n
2752:n
2748:F
2741:n
2724:2
2721:1
2712:|
2706:5
2700:n
2690:|
2658:.
2652:5
2645:1
2641:U
2629:0
2625:U
2618:=
2611:b
2604:,
2598:5
2588:0
2584:U
2575:1
2571:U
2564:=
2557:a
2527:n
2519:b
2516:+
2511:n
2503:a
2500:=
2495:n
2491:U
2477:1
2474:U
2468:0
2465:U
2445:,
2442:a
2436:=
2433:b
2429:,
2423:5
2419:1
2414:=
2398:1
2393:=
2390:a
2359:1
2356:=
2349:b
2343:+
2340:a
2330:0
2327:=
2320:b
2317:+
2314:a
2307:{
2294:b
2290:a
2284:n
2280:U
2273:1
2270:U
2263:0
2260:U
2255:b
2251:a
2231:.
2226:2
2220:n
2216:U
2212:+
2207:1
2201:n
2197:U
2193:=
2181:2
2175:n
2167:b
2164:+
2159:2
2153:n
2145:a
2142:+
2137:1
2131:n
2123:b
2120:+
2115:1
2109:n
2101:a
2098:=
2088:)
2083:2
2077:n
2069:+
2064:1
2058:n
2050:(
2047:b
2044:+
2041:)
2036:2
2030:n
2022:+
2017:1
2011:n
2003:(
2000:a
1997:=
1985:n
1977:b
1974:+
1969:n
1961:a
1958:=
1949:n
1945:U
1914:n
1906:b
1903:+
1898:n
1890:a
1887:=
1882:n
1878:U
1865:b
1861:a
1841:.
1836:2
1830:n
1822:+
1817:1
1811:n
1803:=
1794:n
1782:,
1777:2
1771:n
1763:+
1758:1
1752:n
1744:=
1735:n
1714:Ï
1710:Ï
1696:,
1691:2
1685:n
1681:x
1677:+
1672:1
1666:n
1662:x
1658:=
1653:n
1649:x
1628:1
1625:+
1622:x
1619:=
1614:2
1610:x
1599:Ï
1595:Ï
1579:.
1573:1
1564:2
1557:n
1550:)
1540:(
1532:n
1521:=
1515:5
1508:n
1501:)
1491:(
1483:n
1472:=
1467:n
1463:F
1438:1
1424:=
1399:.
1375:1
1367:=
1358:1
1355:=
1350:2
1344:5
1336:1
1330:=
1311:Ï
1276:2
1270:5
1265:+
1262:1
1256:=
1229:,
1223:5
1216:n
1203:n
1192:=
1173:n
1160:n
1149:=
1144:n
1140:F
1077:F
1051:n
1047:n
1041:n
1034:n
1029:n
989:(
940:.
913:m
909:F
902:m
898:F
890:m
886:F
881:m
871:(
859:m
855:F
850:m
833:5
830:F
824:6
821:F
815:7
812:F
714:F
704:F
694:F
684:F
674:F
664:F
654:F
644:F
634:F
624:F
617:9
614:F
607:8
604:F
597:7
594:F
587:6
584:F
577:5
574:F
567:4
564:F
557:3
554:F
547:2
544:F
537:1
534:F
527:0
524:F
510:n
508:F
498:n
481:2
475:n
471:F
467:+
462:1
456:n
452:F
448:=
443:n
439:F
418:,
415:1
412:=
407:2
403:F
399:=
394:1
390:F
369:0
366:=
361:0
357:F
341:n
324:2
318:n
314:F
310:+
305:1
299:n
295:F
291:=
286:n
282:F
261:,
258:1
255:=
250:1
246:F
241:,
238:0
235:=
230:0
226:F
176:n
172:n
168:n
55:n
53:F
34:.
20:)
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