1013:
1322:
643:
In this numeral system, any substring "100" can be replaced by "011" and vice versa due to the definition of the
Fibonacci numbers. Continual application of these rules will translate form the maximal to the minimal, and vice versa. The fact that any number (greater than 1) can be represented with
467:
Just as the powers of two form a complete sequence due to the binary numeral system, in fact any complete sequence can be used to encode integers as bit strings. The rightmost bit position is assigned to the first, smallest member of the sequence; the next rightmost to the next member; and so on.
255:
51:= 1 + 4 + 32). This sequence is minimal, since no value can be removed from it without making some natural numbers impossible to represent. Simple examples of sequences that are not complete include the
371:
47:, is a complete sequence; given any natural number, we can choose the values corresponding to the 1 bits in its binary representation and sum them to obtain that number (e.g. 37 = 100101
147:
307:
188:
1204:
644:
a terminal 0 means that it is always possible to add 1, and given that, for 1 and 2 can be represented in
Fibonacci coding, completeness follows by
847:
1194:
1287:
633:
619:
440:
395:
1128:
194:
313:
773:
622:). By dropping the trailing one, the coding for 17 above occurs as the 16th term of A104326. The minimal form will never use F
1138:
450:, as well as the Fibonacci numbers with any one number removed. This follows from the identity that the sum of the first
419:
1302:
1133:
893:
840:
1282:
1292:
1184:
1174:
91:
737:
1297:
1199:
833:
726:
S. S. Pillai, "An arithmetical function concerning primes", Annamalai
University Journal (1930), pp. 159–167.
1325:
626:
and will always have a trailing zero. The full coding without the trailing zero can be found at (sequence
1307:
612:
and will always have a trailing one. The full coding without the trailing one can be found at (sequence
1346:
1189:
1179:
1169:
1159:
1274:
1096:
637:
423:
388:
However there are complete sequences that do not satisfy this corollary, for example (sequence
936:
883:
645:
278:
159:
1143:
888:
480:
system, based on the
Fibonacci sequence, the number 17 can be encoded in six different ways:
44:
1254:
1091:
860:
758:
682:
Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., 1985, pp.123-128.
657:
56:
52:
8:
1234:
1101:
777:
433:
which has 1 as the first term and contains all other powers of 2 as a subset. (sequence
1164:
1075:
1060:
1032:
1012:
951:
709:
811:
1264:
1065:
1037:
991:
981:
961:
808:
447:
36:
can be expressed as a sum of values in the sequence, using each value at most once.
1249:
1070:
996:
986:
966:
868:
701:
601:
430:
1027:
956:
754:
745:
468:
Bits set to 1 are included in the sum. These representations may not be unique.
1259:
1244:
1239:
918:
903:
25:
1340:
1224:
898:
40:
1229:
971:
913:
415:
399:
976:
923:
17:
825:
713:
908:
816:
705:
856:
21:
33:
692:
Brown, J. L. (1961). "Note on
Complete Sequences of Integers".
250:{\displaystyle s_{k-1}\geq a_{k}-1{\text{ for all }}k\geq 1}
628:
614:
435:
390:
366:{\displaystyle 2a_{k}\geq a_{k+1}{\text{ for all }}k\geq 0}
76:
is in non-decreasing order, and define the partial sums of
806:
55:, since adding even numbers produces only even numbers—no
1205:
1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)
1195:
1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)
316:
281:
197:
162:
94:
365:
301:
249:
182:
141:
774:"The main operations of the Fibonacci arithmetic"
1338:
67:Without loss of generality, assume the sequence
62:
841:
458: + 2)nd Fibonacci number minus 1.
414:The sequence of the number 1 followed by the
1288:Hypergeometric function of a matrix argument
678:
676:
674:
672:
600:= 13 + 3 + 1 = 17, minimal form, as used in
398:), consisting of the number 1 and the first
1144:1 + 1/2 + 1/3 + ... (Riemann zeta function)
405:
848:
834:
735:
142:{\displaystyle s_{n}=\sum _{m=0}^{n}a_{m}}
1200:1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)
669:
298:
179:
855:
608:The maximal form above will always use F
507:= 8 + 5 + 2 + 1 + 1 = 17, maximal form)
1339:
765:
260:are both necessary and sufficient for
829:
807:
691:
272:A corollary to the above states that
794:. Originally accessed: 27 July 2010.
792:Museum of Harmony and Golden Section
43:(1, 2, 4, 8, ...), the basis of the
1165:1 − 1 + 1 − 1 + ⋯ (Grandi's series)
471:
13:
14:
1358:
1283:Generalized hypergeometric series
800:
694:The American Mathematical Monthly
1321:
1320:
1293:Lauricella hypergeometric series
1011:
410:The complete sequences include:
1303:Riemann's differential equation
636:). This coding is known as the
462:
422:and others); this follows from
729:
720:
685:
1:
1298:Modular hypergeometric series
1139:1/4 + 1/16 + 1/64 + 1/256 + ⋯
663:
39:For example, the sequence of
7:
1308:Theta hypergeometric series
651:
385:to be a complete sequence.
269:to be a complete sequence.
63:Conditions for completeness
10:
1363:
1190:Infinite arithmetic series
1134:1/2 + 1/4 + 1/8 + 1/16 + ⋯
1129:1/2 − 1/4 + 1/8 − 1/16 + ⋯
736:Srinivasan, A. K. (1948),
454:Fibonacci numbers is the (
1316:
1273:
1217:
1152:
1121:
1114:
1084:
1053:
1046:
1020:
1009:
932:
876:
867:
638:Zeckendorf representation
302:{\displaystyle a_{0}=1\,}
183:{\displaystyle a_{0}=1\,}
406:Other complete sequences
1021:Properties of sequences
402:after each power of 2.
884:Arithmetic progression
570:= 13 + 2 + 1 + 1 = 17)
367:
303:
251:
184:
143:
128:
1275:Hypergeometric series
889:Geometric progression
551:= 8 + 5 + 3 + 1 = 17)
529:= 8 + 5 + 3 + 1 = 17)
368:
304:
252:
185:
144:
108:
45:binary numeral system
1255:Trigonometric series
1047:Properties of series
894:Harmonic progression
658:Ostrowski numeration
478:Fibonacci arithmetic
476:For example, in the
424:Bertrand's postulate
314:
279:
195:
160:
153:Then the conditions
92:
1235:Formal power series
812:"Complete Sequence"
780:on January 24, 2013
738:"Practical numbers"
376:are sufficient for
351: for all
235: for all
1033:Monotonic function
952:Fibonacci sequence
809:Weisstein, Eric W.
585:= 13 + 3 + 1 = 17)
363:
299:
247:
180:
139:
32:if every positive
1347:Integer sequences
1334:
1333:
1265:Generating series
1213:
1212:
1185:1 − 2 + 4 − 8 + ⋯
1180:1 + 2 + 4 + 8 + ⋯
1175:1 − 2 + 3 − 4 + ⋯
1170:1 + 2 + 3 + 4 + ⋯
1160:1 + 1 + 1 + 1 + ⋯
1110:
1109:
1038:Periodic sequence
1007:
1006:
992:Triangular number
982:Pentagonal number
962:Heptagonal number
947:Complete sequence
869:Integer sequences
771:Stakhov, Alexey.
448:Fibonacci numbers
431:practical numbers
352:
236:
30:complete sequence
1354:
1324:
1323:
1250:Dirichlet series
1119:
1118:
1051:
1050:
1015:
987:Polygonal number
967:Hexagonal number
940:
874:
873:
850:
843:
836:
827:
826:
822:
821:
795:
789:
787:
785:
776:. Archived from
769:
763:
761:
742:
733:
727:
724:
718:
717:
689:
683:
680:
631:
617:
602:Fibonacci coding
534:
512:
486:
472:Fibonacci coding
438:
429:The sequence of
393:
372:
370:
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364:
353:
350:
348:
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329:
328:
308:
306:
305:
300:
291:
290:
256:
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237:
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226:
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213:
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189:
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104:
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1335:
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1218:Kinds of series
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1115:Explicit series
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1042:
1028:Cauchy sequence
1016:
1003:
957:Figurate number
934:
928:
919:Powers of three
863:
854:
803:
798:
783:
781:
772:
770:
766:
746:Current Science
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706:10.2307/2311150
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408:
389:
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233:
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158:
157:
133:
129:
123:
112:
99:
95:
93:
90:
89:
84:
75:
65:
59:can be formed.
50:
26:natural numbers
12:
11:
5:
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1260:Fourier series
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1245:Puiseux series
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1240:Laurent series
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1048:
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1024:
1022:
1018:
1017:
1010:
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1005:
1004:
1002:
1001:
1000:
999:
989:
984:
979:
974:
969:
964:
959:
954:
949:
943:
941:
930:
929:
927:
926:
921:
916:
911:
906:
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801:External links
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700:(6): 557–560.
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1225:Taylor series
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922:
920:
917:
915:
914:Powers of two
912:
910:
907:
905:
902:
900:
899:Square number
897:
895:
892:
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885:
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804:
793:
784:September 11,
779:
775:
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703:
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673:
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641:
639:
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621:
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587:
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531:
509:
483:
482:
481:
479:
469:
457:
453:
449:
445:
442:
437:
432:
428:
425:
421:
417:
416:prime numbers
413:
412:
411:
403:
401:
397:
392:
386:
383:
379:
360:
357:
354:
344:
341:
338:
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321:
317:
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295:
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283:
275:
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263:
244:
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227:
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214:
209:
206:
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199:
191:
176:
173:
168:
164:
156:
155:
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134:
130:
124:
119:
116:
113:
109:
105:
100:
96:
88:
87:
86:
83:
79:
74:
70:
60:
58:
54:
46:
42:
41:powers of two
37:
35:
31:
27:
23:
19:
1230:Power series
972:Lucas number
946:
924:Powers of 10
904:Cubic number
815:
791:
782:. Retrieved
778:the original
767:
750:
744:
731:
722:
697:
693:
687:
642:
607:
477:
475:
466:
463:Applications
455:
451:
420:S. S. Pillai
418:(studied by
409:
387:
381:
377:
375:
271:
265:
261:
259:
152:
81:
77:
72:
68:
66:
53:even numbers
38:
29:
28:is called a
15:
1097:Conditional
1085:Convergence
1076:Telescoping
1061:Alternating
977:Pell number
753:: 179–180,
18:mathematics
1122:Convergent
1066:Convergent
664:References
588:1001010 (F
573:1001001 (F
554:1000111 (F
57:odd number
1153:Divergent
1071:Divergent
933:Advanced
909:Factorial
857:Sequences
817:MathWorld
646:induction
535:111010 (F
513:111001 (F
487:110111 (F
358:≥
331:≥
242:≥
228:−
215:≥
207:−
110:∑
1341:Category
1326:Category
1092:Absolute
652:See also
22:sequence
1102:Uniform
759:0027799
714:2311150
632:in the
629:A014417
618:in the
615:A104326
533:
511:
485:
439:in the
436:A005153
394:in the
391:A203074
34:integer
1054:Series
861:series
757:
712:
997:array
877:Basic
741:(PDF)
710:JSTOR
400:prime
937:list
859:and
786:2016
634:OEIS
620:OEIS
446:The
441:OEIS
396:OEIS
85:as:
20:, a
702:doi
596:+ F
592:+ F
581:+ F
577:+ F
566:+ F
562:+ F
558:+ F
547:+ F
543:+ F
539:+ F
525:+ F
521:+ F
517:+ F
503:+ F
499:+ F
495:+ F
491:+ F
24:of
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1343::
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704::
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598:2
594:4
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579:4
575:7
568:1
564:2
560:3
556:7
549:2
545:4
541:5
537:6
527:1
523:4
519:5
515:6
505:1
501:2
497:3
493:5
489:6
456:n
452:n
443:)
426:.
382:n
378:a
361:0
355:k
345:1
342:+
339:k
335:a
326:k
322:a
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296:1
293:=
288:0
284:a
266:n
262:a
245:1
239:k
231:1
223:k
219:a
210:1
204:k
200:s
177:1
174:=
169:0
165:a
149:.
135:m
131:a
125:n
120:0
117:=
114:m
106:=
101:n
97:s
82:n
78:a
73:n
69:a
49:2
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