2145:(The division is not displayed as a fraction for emphasizing that it must be computed after the multiplication, for not introducing fractional numbers). This recurrence is widely used in computers because it does not require to build a table as does the bi-dimensional recurrence, and does involve very large integers as does the formula with factorials (if one uses
7354:. However, it requires a sorted vector. It will first check if the element is at the middle of the vector. If not, then it will check if the middle element is greater or lesser than the sought element. At this point, half of the vector can be discarded, and the algorithm can be run again on the other half. The number of comparisons will be given by
8023:
Recurrence relations, especially linear recurrence relations, are used extensively in both theoretical and empirical economics. In particular, in macroeconomics one might develop a model of various broad sectors of the economy (the financial sector, the goods sector, the labor market, etc.) in which
3910:
4089:
4802:
4627:
4466:
5766:
774:
3220:
8000:
7871:
7017:
is used either directly to model population growth, or as a starting point for more detailed models of population dynamics. In this context, coupled difference equations are often used to model the interaction of two or more
6854:
3070:
3468:
Difference equations resemble differential equations, and this resemblance is often used to mimic methods for solving differentiable equations to apply to solving difference equations, and therefore recurrence relations.
3456:
As it is equivalent for a sequence to satisfy a recurrence relation or to be the solution of a difference equation, the two terms "recurrence relation" and "difference equation" are sometimes used interchangeably. See
513:
2778:
1814:
5637:
4283:
3733:
4181:
3918:
921:
6980:
3652:
4633:
2095:
6669:
3365:
2875:
2940:
1985:
7628:
824:
6365:
5397:
5212:
5031:
1891:
559:
7103:
4898:
3491:
Single-variable or one-dimensional recurrence relations are about sequences (i.e. functions defined on one-dimensional grids). Multi-variable or n-dimensional recurrence relations are about
2218:
2658:
7188:
4472:
2437:
4289:
2353:
3448:
3727:
1082:
1603:
6471:
6244:
1242:
261:
6068:
1485:
1431:
1377:
7327:
A naive algorithm will search from left to right, one element at a time. The worst possible scenario is when the required element is the last, so the number of comparisons is
1655:
7498:
7443:
5089:
5272:
2143:
2516:
599:
2696:
2546:
1542:
5651:
2486:
2463:
2269:
1318:
655:
7745:
7388:
3081:
2566:
1284:
6300:
4924:
2596:
7705:
7658:
7245:
7218:
6501:
6180:
6133:
6106:
5975:
5478:
5451:
5424:
1158:
956:
626:
5885:
1131:
1108:
308:
7725:
7678:
7545:
7345:
7322:
7265:
6696:
6528:
6408:
6388:
6274:
6153:
5995:
5948:
5928:
5908:
5511:
3529:
3509:
1695:
1675:
990:
395:
356:
332:
281:
192:
157:
133:
113:
93:
73:
49:
432:
as a function of the preceding ones. More precisely, in the case where only the immediately preceding element is involved, a recurrence relation has the form
7877:
6249:
A nonlinear recurrence could have multiple fixed points, in which case some fixed points may be locally stable and others locally unstable; for continuous
6530:
stays in a bounded region but never converges to a fixed point or an attracting cycle; any fixed points or cycles of the equation are unstable. See also
7753:
6704:
8617:
index to a few thousand examples of linear recurrences, sorted by order (number of terms) and signature (vector of values of the constant coefficients)
2948:
8477:
8408:
438:
5290:
2711:
1703:
5519:
3905:{\displaystyle {\frac {a_{n+1}}{\prod _{k=0}^{n}f_{k}}}-{\frac {f_{n}a_{n}}{\prod _{k=0}^{n}f_{k}}}={\frac {g_{n}}{\prod _{k=0}^{n}f_{k}}}}
3549:
284:
4192:
8427:
4084:{\displaystyle {\frac {a_{n+1}}{\prod _{k=0}^{n}f_{k}}}-{\frac {a_{n}}{\prod _{k=0}^{n-1}f_{k}}}={\frac {g_{n}}{\prod _{k=0}^{n}f_{k}}}}
8192:
3280:
of the other, and the sequences that are solution of the difference equation are exactly those that satisfies the recurrence relation.
6554:
4100:
857:
6865:
4797:{\displaystyle a_{n}=\left(\prod _{k=0}^{n-1}f_{k}\right)\left(A_{0}+\sum _{m=0}^{n-1}{\frac {g_{m}}{\prod _{k=0}^{m}f_{k}}}\right)}
7512:, recurrence relations can model feedback in a system, where outputs at one time become inputs for future time. They thus arise in
3565:
8072:
1996:
6302:. Such a cycle is stable, meaning that it attracts a set of initial conditions of positive measure, if the composite function
3289:
2786:
8149:
2890:
1907:
7553:
6985:
Systems of linear first order differential equations can be discretized exactly analytically using the methods shown in the
782:
6308:
5318:
5104:
201:
4956:
4930:
with variable coefficients; the sum becomes an integral, and the product becomes the exponential function of an integral.
8168:
6566:
1822:
159:
numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation.
524:
8024:
some agents' actions depend on lagged variables. The model would then be solved for current values of key variables (
7036:
4810:
4622:{\displaystyle {\frac {a_{n}}{\prod _{k=0}^{n-1}f_{k}}}=A_{0}+\sum _{m=0}^{n-1}{\frac {g_{m}}{\prod _{k=0}^{m}f_{k}}}}
8465:
8437:
8397:
8381:
8290:
8254:
8102:
4939:
4461:{\displaystyle \sum _{m=0}^{n-1}(A_{m+1}-A_{m})=A_{n}-A_{0}=\sum _{m=0}^{n-1}{\frac {g_{m}}{\prod _{k=0}^{m}f_{k}}}}
2606:
2148:
7109:
2371:
5278:
2281:
6551:
3376:
3242:
3663:
1013:
8635:
8582:
5453:
which itself evolves linearly. Then standard methods can be used to solve the linear difference equation in
3559:
Moreover, for the general first-order non-homogeneous linear recurrence relation with variable coefficients:
1557:
8526:
Wang, Xiang-Sheng; Wong, Roderick (2012). "Asymptotics of orthogonal polynomials via recurrence relations".
8077:
8062:
8018:
7298:
5310:
4927:
3458:
1893:. Using this formula to compute the values of all binomial coefficients generates an infinite array called
1182:
8488:
1897:. The same values can also be computed directly by a different formula that is not a recurrence, but uses
8577:
8572:
1176:
relation with constant coefficients (see below). The
Fibonacci sequence is defined using the recurrence
5298:
8372:
8067:
7270:
7023:
6416:
6192:
6015:
5792:
3462:
1437:
1383:
1329:
635:
It is easy to modify the definition for getting sequences starting from the term of index 1 or higher.
3554:
1619:
8097:
7513:
7509:
7455:
7394:
5643:
5042:
8414:
5223:
8092:
2103:
2491:
7351:
5761:{\displaystyle \lambda ^{d}-c_{1}\lambda ^{d-1}-c_{2}\lambda ^{d-2}-\cdots -c_{d}\lambda ^{0}=0.}
769:{\displaystyle u_{n}=\varphi (n,u_{n-1},u_{n-2},\ldots ,u_{n-k})\quad {\text{for}}\quad n\geq k,}
7747:
controls how much of the delayed signal is fed back into the output. From this we can see that
3215:{\displaystyle a_{n+k}=a_{n}+{k \choose 1}\Delta a_{n}+\cdots +{k \choose k}\Delta ^{k}(a_{n}).}
571:
287:, because the coefficients of the linear function (1 and 1) are constants that do not depend on
8640:
8457:
8107:
8052:
7290:
5294:
4943:
2675:
2525:
2245:
2237:
1508:
401:
311:
2468:
2445:
2251:
1290:
7730:
7360:
7304:
A simple example is the time an algorithm takes to find an element in an ordered vector with
6558:
6535:
5779:(i.e., the roots of the characteristic equation), whether real or complex, are all less than
3483:
for a unification of the theory of difference equations with that of differential equations.
2551:
1256:
8449:
6279:
4903:
2571:
8359:
8014:
7683:
7636:
7223:
7196:
6479:
6158:
6111:
6084:
5953:
5456:
5429:
5402:
5282:
1894:
1614:
1136:
932:
604:
374:
335:
8:
8087:
7003:
6183:
6078:
5780:
5775:, meaning that the iterates converge asymptotically to a fixed value, if and only if the
3480:
2272:
1545:
8594:
5803:
1113:
1090:
290:
8553:
8535:
8333:
8279:
8240:
8082:
8042:
7995:{\displaystyle y_{t}=(1-\alpha )x_{t}+(\alpha -\alpha ^{2})x_{t-T}+\alpha ^{2}y_{t-2T}}
7710:
7663:
7530:
7330:
7307:
7250:
7002:
Some of the best-known difference equations have their origins in the attempt to model
6681:
6513:
6393:
6373:
6259:
6138:
5980:
5933:
5913:
5893:
5496:
5286:
3514:
3494:
3472:
1680:
1660:
1502:
975:
380:
367:
341:
317:
266:
177:
142:
118:
98:
78:
58:
34:
8591:
8461:
8450:
8433:
8393:
8377:
8286:
8274:
8250:
8236:
8186:
8145:
8047:
6675:
3476:
2359:
1549:
1248:
1173:
359:
283:
is two and the linear function merely adds the two previous terms. This example is a
8557:
8630:
8545:
8517:
8363:
8355:
7007:
6074:
5772:
4947:
3555:
Solving first-order non-homogeneous recurrence relations with variable coefficients
3277:
1169:
405:
195:
7866:{\displaystyle y_{t}=(1-\alpha )x_{t}+\alpha ((1-\alpha )x_{t-T}+\alpha y_{t-2T})}
6849:{\displaystyle y_{0}=y(t_{0}),\ \ y_{1}=y(t_{0}+h),\ \ y_{2}=y(t_{0}+2h),\ \dots }
1990:
The binomial coefficients can also be computed with a uni-dimensional recurrence:
8270:
8244:
8141:
7449:
5095:
3543:
3065:{\displaystyle \Delta ^{k}a_{n}=\sum _{t=0}^{k}(-1)^{t}{\binom {k}{t}}a_{n+k-t}.}
171:
8161:
6557:, one typically encounters a recurrence relation. For example, when solving the
8507:
8478:"Using generating functions to solve linear inhomogeneous recurrence equations"
8367:
7517:
6986:
5998:
413:
8549:
3465:
for example of uses of "difference equation" instead of "recurrence relation"
8624:
8609:
8025:
363:
310:
For these recurrences, one can express the general term of the sequence as a
8423:
8232:
7277:. These and other difference equations are particularly suited to modeling
7014:
6531:
6507:
4938:
Many homogeneous linear recurrence relations may be solved by means of the
1613:
A simple example of a multidimensional recurrence relation is given by the
1004:
55:
of numbers is equal to some combination of the previous terms. Often, only
5289:. For these specific recurrence equations algorithms are known which find
4933:
3260:
The two above relations allow transforming a recurrence relation of order
1505:, which involves powers of the two roots of the characteristic polynomial
7524:
5776:
3453:
in the sense that the two equations are satisfied by the same sequences.
20:
8328:
Fillmore, Jay P.; Marx, Morris L. (1968). "Linear recursive sequences".
508:{\displaystyle u_{n}=\varphi (n,u_{n-1})\quad {\text{for}}\quad n>0,}
8337:
7019:
3254:
2773:{\displaystyle \Delta ^{2}a=(\Delta \circ \Delta )a=\Delta (\Delta a).}
75:
previous terms of the sequence appear in the equation, for a parameter
7297:
is designed so that it will break a problem into smaller subproblems (
1809:{\displaystyle {\binom {n}{k}}={\binom {n-1}{k-1}}+{\binom {n-1}{k}},}
8599:
8057:
7294:
7278:
5786:
5632:{\displaystyle a_{n}=c_{1}a_{n-1}+c_{2}a_{n-2}+\cdots +c_{d}a_{n-d},}
1898:
848:
8346:
5304:
4278:{\displaystyle A_{n+1}-A_{n}={\frac {g_{n}}{\prod _{k=0}^{n}f_{k}}}}
3241:
first differences of a sequence or a function, in the same way as a
2365:
When using the index notation for sequences, the definition becomes
7027:
6539:
2601:
2241:
429:
52:
28:
8540:
6256:
A nonlinear recurrence relation could also have a cycle of period
568:
is a set to which the elements of a sequence must belong. For any
7274:
7010:
were once used as a model for the growth of a rabbit population.
6004:
1501:
The recurrence can be solved by methods described below yielding
8032:, etc.) in terms of past and current values of other variables.
8485:
Proc. Int. Conf. Simulation, Modelling and
Optimization, SMO'06
5488:
4176:{\displaystyle A_{n}={\frac {a_{n}}{\prod _{k=0}^{n-1}f_{k}}},}
2223:
916:{\displaystyle n!=n\cdot (n-1)!\quad {\text{for}}\quad n>0,}
6975:{\displaystyle \,y_{n+1}=y_{n}+hf(t_{n},y_{n}),t_{n}=t_{0}+nh}
4942:. Special cases of these lead to recurrence relations for the
3544:
Solving linear recurrence relations with constant coefficients
8388:
Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1994).
409:
8589:
3647:{\displaystyle a_{n+1}=f_{n}a_{n}+g_{n},\qquad f_{n}\neq 0,}
8614:
7301:), its running time is described by a recurrence relation.
7289:
Recurrence relations are also of fundamental importance in
2090:{\displaystyle {\binom {n}{k}}={\binom {n}{k-1}}(n-k+1)/k,}
366:
whose coefficients satisfy such a recurrence relation (see
194:
previous terms. A famous example is the recurrence for the
8132:
Jacobson, Nathan, Basic
Algebra 2 (2nd ed.), § 0.4. pg 16.
2220:
all involved integers are smaller than the final result).
1697:
elements. They can be computed by the recurrence relation
1493:
We obtain the sequence of
Fibonacci numbers, which begins
8029:
6545:
6410:
times is locally stable according to the same criterion:
6253:
two adjacent fixed points cannot both be locally stable.
5977:
if and only if all eigenvalues of the transition matrix
5315:
A first order rational difference equation has the form
5283:
linear difference equations with polynomial coefficients
3360:{\displaystyle 3\Delta ^{2}a_{n}+2\Delta a_{n}+7a_{n}=0}
2870:{\displaystyle \Delta ^{2}a_{n}=a_{n+2}-2a_{n+1}+a_{n}.}
400:
The concept of a recurrence relation can be extended to
8390:
Concrete
Mathematics: A Foundation for Computer Science
7273:
are a form of recurrence relation important to spatial
4934:
Solving general homogeneous linear recurrence relations
2935:{\displaystyle \Delta ^{k}=\Delta \circ \Delta ^{k-1},}
1980:{\displaystyle {\binom {n}{k}}={\frac {n!}{k!(n-k)!}}.}
358:
are also important, because many common elementary and
8508:"Difference and Functional Equations: Exact Solutions"
7623:{\displaystyle y_{t}=(1-\alpha )x_{t}+\alpha y_{t-T},}
5342:
2151:
2106:
1854:
1827:
1624:
819:{\displaystyle \varphi :\mathbb {N} \times X^{k}\to X}
7880:
7756:
7733:
7713:
7686:
7666:
7639:
7556:
7533:
7458:
7397:
7363:
7333:
7310:
7253:
7226:
7199:
7112:
7039:
6868:
6707:
6684:
6569:
6516:
6482:
6419:
6396:
6376:
6360:{\displaystyle g(x):=f\circ f\circ \cdots \circ f(x)}
6311:
6282:
6262:
6195:
6161:
6141:
6114:
6087:
6018:
5983:
5956:
5936:
5916:
5896:
5806:
5654:
5522:
5499:
5459:
5432:
5405:
5392:{\displaystyle w_{t+1}={\tfrac {aw_{t}+b}{cw_{t}+d}}}
5321:
5226:
5207:{\displaystyle (b-n)M_{n-1}+(2n-b+z)M_{n}-nM_{n+1}=0}
5107:
5045:
4959:
4906:
4813:
4636:
4475:
4292:
4195:
4103:
3921:
3736:
3666:
3568:
3517:
3497:
3379:
3292:
3084:
2951:
2893:
2789:
2714:
2678:
2609:
2574:
2554:
2528:
2494:
2471:
2448:
2374:
2284:
2254:
1999:
1910:
1825:
1706:
1683:
1663:
1622:
1560:
1511:
1440:
1386:
1332:
1293:
1259:
1185:
1139:
1116:
1110:
The behavior of the sequence depends dramatically on
1093:
1016:
978:
935:
860:
785:
658:
607:
574:
527:
441:
383:
344:
320:
293:
269:
204:
180:
145:
121:
101:
81:
61:
37:
8387:
5950:
converges asymptotically to the steady state vector
5026:{\displaystyle J_{n+1}={\frac {2n}{z}}J_{n}-J_{n-1}}
830:
consecutive elements of the sequence. In this case,
8376:, Second Edition. MIT Press and McGraw-Hill, 1990.
8231:
6664:{\displaystyle y'(t)=f(t,y(t)),\ \ y(t_{0})=y_{0},}
1901:, multiplication and division, not just additions:
1886:{\displaystyle {\tbinom {n}{0}}={\tbinom {n}{n}}=1}
834:initial values are needed for defining a sequence.
8278:
7994:
7865:
7739:
7719:
7699:
7672:
7652:
7622:
7539:
7523:For example, the equation for a "feedforward" IIR
7492:
7437:
7382:
7339:
7316:
7259:
7239:
7212:
7182:
7097:
6974:
6848:
6690:
6663:
6522:
6495:
6465:
6402:
6382:
6359:
6294:
6268:
6238:
6174:
6147:
6127:
6100:
6062:
5989:
5969:
5942:
5922:
5902:
5879:
5787:Stability of linear first-order matrix recurrences
5760:
5631:
5505:
5472:
5445:
5426:as a nonlinear transformation of another variable
5418:
5391:
5266:
5206:
5083:
5025:
4918:
4892:
4796:
4621:
4460:
4277:
4175:
4083:
3904:
3721:
3646:
3523:
3503:
3442:
3359:
3268:, and, conversely, a difference equation of order
3214:
3064:
2934:
2869:
2772:
2690:
2652:
2590:
2560:
2540:
2510:
2480:
2457:
2431:
2347:
2263:
2212:
2137:
2089:
1979:
1885:
1808:
1689:
1669:
1649:
1597:
1536:
1479:
1425:
1371:
1312:
1278:
1236:
1152:
1125:
1102:
1076:
984:
950:
915:
818:
768:
620:
593:
554:{\displaystyle \varphi :\mathbb {N} \times X\to X}
553:
507:
389:
350:
326:
302:
275:
255:
186:
151:
127:
107:
87:
67:
43:
8522:at EqWorld - The World of Mathematical Equations.
8512:at EqWorld - The World of Mathematical Equations.
8421:
8269:
5305:Solving first-order rational difference equations
3177:
3164:
3133:
3120:
3031:
3018:
2049:
2028:
2016:
2003:
1927:
1914:
1797:
1776:
1764:
1735:
1723:
1710:
8622:
7098:{\displaystyle N_{t+1}=\lambda N_{t}e^{-aP_{t}}}
4893:{\displaystyle a_{n+1}=(1+hf_{nh})a_{n}+hg_{nh}}
1323:Explicitly, the recurrence yields the equations
428:is an equation that expresses each element of a
373:Solving a recurrence relation means obtaining a
16:Pattern defining an infinite sequence of numbers
2213:{\textstyle {\binom {n}{k}}={\binom {n}{n-k}},}
336:linear recurrences with polynomial coefficients
8518:"Difference and Functional Equations: Methods"
8332:. Vol. 10, no. 3. pp. 324–353.
8312:An introduction to linear difference equations
6009:Consider the nonlinear first-order recurrence
6005:Stability of nonlinear first-order recurrences
5797:In the first-order matrix difference equation
2653:{\displaystyle a=(a_{n})_{n\in \mathbb {N} },}
963:linear recurrence with polynomial coefficients
8220:An Introduction to the Analysis of Algorithms
7183:{\displaystyle P_{t+1}=N_{t}(1-e^{-aP_{t}}),}
2432:{\displaystyle (\Delta a)_{n}=a_{n+1}-a_{n}.}
2224:Difference operator and difference equations
2201:
2180:
2168:
2155:
2123:
2110:
1870:
1857:
1843:
1830:
1640:
1627:
1168:The recurrence of order two satisfied by the
8327:
7503:
5489:Stability of linear higher-order recurrences
5399:. Such an equation can be solved by writing
3550:Linear recurrence with constant coefficients
1497:0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
285:linear recurrence with constant coefficients
139:of the relation. If the values of the first
8429:Difference Equations: From Rabbits to Chaos
1003:An example of a recurrence relation is the
965:of order 1, with the simple polynomial (in
8475:
8456:(Fifth ed.). Prentice Hall. pp.
8309:
3486:
2348:{\displaystyle (\Delta f)(x)=f(x+1)-f(x).}
1172:is the canonical example of a homogeneous
8539:
8384:. Chapter 4: Recurrences, pp. 62–90.
8344:
8285:(Second ed.). Cambridge: MIT Press.
6869:
3657:there is also a nice method to solve it:
3511:-dimensional grids. Functions defined on
3443:{\displaystyle 3a_{n+2}=4a_{n+1}-8a_{n},}
3370:is equivalent to the recurrence relation
2641:
1133:but is stable when the initial condition
793:
535:
8525:
8348:Linear Recursion and Fibonacci Sequences
3722:{\displaystyle a_{n+1}-f_{n}a_{n}=g_{n}}
2518:must be understood as the term of index
1608:
1077:{\displaystyle x_{n+1}=rx_{n}(1-x_{n}),}
8447:
8249:. Cambridge: Harvard University Press.
8073:Master theorem (analysis of algorithms)
6997:
5281:. Sequences which are the solutions of
1598:{\displaystyle {\frac {t}{1-t-t^{2}}}.}
8623:
8452:Mathematics for Economics and Business
8406:
8318:
8246:Recursive Methods in Economic Dynamics
8191:: CS1 maint: archived copy as title (
6546:Relationship to differential equations
3479:relate to differential equations. See
3075:This relation can be inverted, giving
851:is defined by the recurrence relation
601:, this defines a unique sequence with
8590:
4926:, we get the formula for first order
3283:For example, the difference equation
2248:to functions. It is commonly denoted
1237:{\displaystyle F_{n}=F_{n-1}+F_{n-2}}
256:{\displaystyle F_{n}=F_{n-1}+F_{n-2}}
8515:
8505:
3264:into a difference equation of order
1657:, which count the ways of selecting
1163:
638:This defines recurrence relation of
7284:
2244:to sequences, and, more generally,
13:
8487:. pp. 399–404. Archived from
8472:Chapter 9.1: Difference Equations.
8144:, Sui Sun Cheng, CRC Press, 2003,
6510:recurrence relation, the variable
6108:from points sufficiently close to
5997:(whether real or complex) have an
3475:relate to difference equations as
3322:
3297:
3272:into recurrence relation of order
3184:
3168:
3139:
3124:
3022:
2953:
2914:
2907:
2895:
2791:
2758:
2752:
2740:
2734:
2716:
2679:
2555:
2529:
2495:
2472:
2449:
2378:
2288:
2255:
2184:
2159:
2114:
2032:
2007:
1918:
1861:
1834:
1780:
1739:
1714:
1631:
14:
8652:
8565:
8103:Integration by reduction formulae
6466:{\displaystyle |g'(x^{*})|<1,}
6239:{\displaystyle |f'(x^{*})|<1.}
6063:{\displaystyle x_{n}=f(x_{n-1}).}
4940:generalized hypergeometric series
3237:is an equation that involves the
1480:{\displaystyle F_{4}=F_{3}+F_{2}}
1426:{\displaystyle F_{3}=F_{2}+F_{1}}
1372:{\displaystyle F_{2}=F_{1}+F_{0}}
628:as its first element, called the
8476:Minh, Tang; Van To, Tan (2006).
8410:Applied Econometric Times Series
3531:-grids can also be studied with
2780:A simple computation shows that
1650:{\displaystyle {\tbinom {n}{k}}}
8303:
8174:from the original on 2010-07-05
7493:{\displaystyle O(\log _{2}(n))}
7438:{\displaystyle c_{n}=1+c_{n/2}}
6992:
5493:The linear recurrence of order
5279:confluent hypergeometric series
5084:{\displaystyle J_{n}=J_{n}(z),}
4950:. For example, the solution to
3624:
998:
900:
894:
753:
747:
492:
486:
8392:(2 ed.). Addison-Wesley.
8281:Recursive Macroeconomic Theory
8263:
8225:
8212:
8199:
8154:
8135:
8126:
7941:
7922:
7906:
7894:
7860:
7816:
7804:
7801:
7782:
7770:
7582:
7570:
7487:
7484:
7478:
7462:
7324:elements, in the worst case.
7174:
7142:
6934:
6908:
6834:
6812:
6784:
6765:
6737:
6724:
6642:
6629:
6614:
6611:
6605:
6593:
6584:
6578:
6552:ordinary differential equation
6450:
6446:
6433:
6421:
6354:
6348:
6321:
6315:
6226:
6222:
6209:
6197:
6054:
6035:
5874:
5842:
5833:
5807:
5267:{\displaystyle M_{n}=M(n,b;z)}
5261:
5243:
5163:
5142:
5120:
5108:
5075:
5069:
4910:
4858:
4833:
4352:
4320:
3206:
3193:
3006:
2996:
2764:
2755:
2743:
2731:
2630:
2616:
2385:
2375:
2339:
2333:
2324:
2312:
2303:
2297:
2294:
2285:
2138:{\textstyle {\binom {n}{0}}=1}
2073:
2055:
1965:
1953:
1068:
1049:
888:
876:
810:
744:
675:
545:
483:
458:
377:: a non-recursive function of
1:
8114:
7350:A better algorithm is called
6186:in absolute value: that is,
4928:linear differential equations
3276:. Each transformation is the
2358:It is thus a special case of
419:
8413:(3 ed.). Archived from
8310:Batchelder, Paul M. (1967).
8142:Partial difference equations
8119:
8078:Circle points segments proof
8063:Recursion (computer science)
8019:simultaneous equations model
8008:
7220:representing the hosts, and
6698:, one calculates the values
5483:
5311:Rational difference equation
3533:partial difference equations
3459:Rational difference equation
2511:{\displaystyle \Delta a_{n}}
842:
826:is a function that involves
7:
8578:Encyclopedia of Mathematics
8321:Linear difference equations
8319:Miller, Kenneth S. (1968).
8218:R. Sedgewick, F. Flajolet,
8035:
7271:Integrodifference equations
6503:is any point on the cycle.
6001:which is less than 1.
4807:If we apply the formula to
2488:are generally omitted, and
837:
642:. A recurrence relation of
10:
8659:
8373:Introduction to Algorithms
8345:Brousseau, Alfred (1971).
8207:Introduction to Algorithms
8068:Lagged Fibonacci generator
8012:
5793:Matrix difference equation
5790:
5308:
3547:
3538:
3463:Matrix difference equation
2887:is defined recursively as
926:and the initial condition
594:{\displaystyle u_{0}\in X}
8550:10.1142/S0219530512500108
8426:; Robson, Robbie (2005).
8098:Infinite impulse response
7514:infinite impulse response
7510:digital signal processing
7504:Digital signal processing
2691:{\displaystyle \Delta a.}
2541:{\displaystyle \Delta a,}
1677:elements out of a set of
1537:{\displaystyle t^{2}=t+1}
995:as its only coefficient.
8351:. Fibonacci Association.
8093:Combinatorial principles
7030:interaction is given by
2481:{\displaystyle \Delta a}
2458:{\displaystyle \Delta f}
2264:{\displaystyle \Delta ,}
1313:{\displaystyle F_{1}=1.}
961:This is an example of a
170:th term is equated to a
8407:Enders, Walter (2010).
7740:{\displaystyle \alpha }
7383:{\displaystyle c_{1}=1}
7247:the parasites, at time
6155:in the neighborhood of
5644:characteristic equation
3487:From sequences to grids
2568:applied to the element
2561:{\displaystyle \Delta }
2442:The parentheses around
2100:with the initial value
1548:of the sequence is the
1279:{\displaystyle F_{0}=0}
402:multidimensional arrays
95:that is independent of
31:according to which the
8222:, Addison-Wesley, 2013
8108:Mathematical induction
8053:Orthogonal polynomials
7996:
7867:
7741:
7721:
7707:is the output at time
7701:
7674:
7654:
7624:
7541:
7494:
7439:
7384:
7341:
7318:
7291:analysis of algorithms
7261:
7241:
7214:
7184:
7099:
7024:Nicholson–Bailey model
6976:
6850:
6692:
6665:
6524:
6497:
6467:
6404:
6384:
6361:
6296:
6295:{\displaystyle k>1}
6270:
6240:
6176:
6149:
6129:
6102:
6064:
5991:
5971:
5944:
5924:
5910:and transition matrix
5904:
5881:
5762:
5633:
5507:
5474:
5447:
5420:
5393:
5268:
5208:
5085:
5027:
4944:orthogonal polynomials
4920:
4919:{\displaystyle h\to 0}
4894:
4798:
4775:
4741:
4681:
4623:
4605:
4571:
4515:
4462:
4444:
4410:
4319:
4279:
4261:
4177:
4156:
4085:
4067:
4017:
3961:
3906:
3888:
3838:
3776:
3723:
3648:
3525:
3505:
3444:
3361:
3216:
3066:
2995:
2936:
2871:
2774:
2692:
2654:
2592:
2591:{\displaystyle a_{n}.}
2562:
2542:
2512:
2482:
2459:
2433:
2349:
2265:
2214:
2139:
2091:
1981:
1887:
1810:
1691:
1671:
1651:
1599:
1538:
1481:
1427:
1373:
1314:
1280:
1238:
1154:
1127:
1104:
1078:
986:
952:
917:
820:
770:
622:
595:
555:
509:
391:
352:
328:
312:closed-form expression
304:
277:
257:
188:
153:
129:
109:
89:
69:
45:
8595:"Recurrence Equation"
8573:"Recurrence relation"
8448:Jacques, Ian (2006).
8314:. Dover Publications.
7997:
7868:
7742:
7722:
7702:
7700:{\displaystyle y_{t}}
7675:
7660:is the input at time
7655:
7653:{\displaystyle x_{t}}
7625:
7542:
7495:
7440:
7385:
7342:
7319:
7262:
7242:
7240:{\displaystyle P_{t}}
7215:
7213:{\displaystyle N_{t}}
7185:
7100:
6977:
6851:
6693:
6666:
6559:initial value problem
6536:dyadic transformation
6525:
6498:
6496:{\displaystyle x^{*}}
6468:
6405:
6385:
6362:
6297:
6271:
6241:
6177:
6175:{\displaystyle x^{*}}
6150:
6130:
6128:{\displaystyle x^{*}}
6103:
6101:{\displaystyle x^{*}}
6065:
5992:
5972:
5970:{\displaystyle x^{*}}
5945:
5925:
5905:
5882:
5763:
5634:
5508:
5475:
5473:{\displaystyle x_{t}}
5448:
5446:{\displaystyle x_{t}}
5421:
5419:{\displaystyle w_{t}}
5394:
5269:
5209:
5086:
5028:
4921:
4895:
4799:
4755:
4715:
4655:
4624:
4585:
4545:
4489:
4463:
4424:
4384:
4293:
4280:
4241:
4178:
4130:
4086:
4047:
3991:
3941:
3907:
3868:
3818:
3756:
3724:
3649:
3526:
3506:
3445:
3362:
3243:differential equation
3217:
3067:
2975:
2937:
2880:More generally: the
2872:
2775:
2693:
2655:
2593:
2563:
2543:
2513:
2483:
2460:
2434:
2350:
2266:
2215:
2140:
2092:
1982:
1888:
1811:
1692:
1672:
1652:
1615:binomial coefficients
1609:Binomial coefficients
1600:
1539:
1482:
1428:
1374:
1315:
1281:
1239:
1155:
1153:{\displaystyle x_{0}}
1128:
1105:
1087:for a given constant
1079:
987:
953:
951:{\displaystyle 0!=1.}
918:
821:
771:
623:
621:{\displaystyle u_{0}}
596:
564:is a function, where
556:
510:
392:
353:
329:
305:
278:
258:
189:
154:
130:
110:
90:
70:
46:
8636:Recurrence relations
8516:Polyanin, Andrei D.
8506:Polyanin, Andrei D.
8360:Charles E. Leiserson
8237:Lucas, Robert E. Jr.
8015:time series analysis
7878:
7754:
7731:
7711:
7684:
7664:
7637:
7554:
7531:
7456:
7395:
7361:
7331:
7308:
7251:
7224:
7197:
7110:
7037:
6998:Mathematical biology
6866:
6705:
6682:
6567:
6514:
6480:
6417:
6394:
6374:
6309:
6280:
6260:
6193:
6159:
6139:
6112:
6085:
6016:
5981:
5954:
5934:
5914:
5894:
5804:
5652:
5520:
5497:
5457:
5430:
5403:
5319:
5224:
5105:
5043:
4957:
4904:
4811:
4634:
4473:
4290:
4193:
4101:
3919:
3734:
3664:
3566:
3515:
3495:
3377:
3290:
3082:
2949:
2891:
2787:
2712:
2676:
2607:
2572:
2552:
2526:
2492:
2469:
2446:
2372:
2282:
2252:
2149:
2104:
1997:
1908:
1823:
1819:with the base cases
1704:
1681:
1661:
1620:
1558:
1509:
1438:
1384:
1330:
1291:
1257:
1183:
1137:
1114:
1091:
1014:
976:
933:
858:
783:
656:
605:
572:
525:
439:
408:that are indexed by
381:
375:closed-form solution
342:
318:
291:
267:
202:
178:
143:
119:
99:
79:
59:
35:
8241:Prescott, Edward C.
8088:Time scale calculus
8043:Holonomic sequences
7022:. For example, the
7006:. For example, the
7004:population dynamics
6073:This recurrence is
5783:in absolute value.
4900:and take the limit
3481:time scale calculus
3473:Summation equations
3229:difference equation
2273:functional notation
2271:and is defined, in
2232:difference operator
1546:generating function
426:recurrence relation
25:recurrence relation
8592:Weisstein, Eric W.
8275:Sargent, Thomas J.
8205:Cormen, T. et al,
8083:Continued fraction
7992:
7863:
7737:
7717:
7697:
7670:
7650:
7620:
7537:
7490:
7435:
7380:
7337:
7314:
7299:divide and conquer
7257:
7237:
7210:
7180:
7095:
6972:
6859:by the recurrence
6846:
6688:
6661:
6520:
6493:
6463:
6400:
6380:
6357:
6292:
6266:
6236:
6172:
6145:
6135:, if the slope of
6125:
6098:
6077:, meaning that it
6060:
5987:
5967:
5940:
5920:
5900:
5890:with state vector
5880:{\displaystyle =A}
5877:
5771:The recurrence is
5758:
5629:
5503:
5470:
5443:
5416:
5389:
5387:
5264:
5204:
5081:
5023:
4916:
4890:
4794:
4619:
4458:
4275:
4173:
4081:
3902:
3719:
3644:
3521:
3501:
3477:integral equations
3440:
3357:
3212:
3062:
2932:
2867:
2770:
2688:
2650:
2588:
2558:
2538:
2508:
2478:
2455:
2429:
2345:
2261:
2210:
2135:
2087:
1977:
1883:
1875:
1848:
1806:
1687:
1667:
1647:
1645:
1595:
1534:
1477:
1423:
1369:
1310:
1276:
1249:initial conditions
1234:
1150:
1126:{\displaystyle r,}
1123:
1103:{\displaystyle r.}
1100:
1074:
982:
948:
913:
816:
766:
618:
591:
551:
505:
387:
368:holonomic function
348:
324:
303:{\displaystyle n.}
300:
273:
253:
184:
164:linear recurrences
149:
125:
105:
85:
65:
41:
8323:. W. A. Benjamin.
8209:, MIT Press, 2009
8150:978-0-415-29884-1
8048:Iterated function
7720:{\displaystyle t}
7673:{\displaystyle t}
7540:{\displaystyle T}
7452:of which will be
7340:{\displaystyle n}
7317:{\displaystyle n}
7260:{\displaystyle t}
7008:Fibonacci numbers
6842:
6795:
6792:
6748:
6745:
6691:{\displaystyle h}
6625:
6622:
6523:{\displaystyle x}
6403:{\displaystyle k}
6383:{\displaystyle f}
6269:{\displaystyle k}
6148:{\displaystyle f}
6081:to a fixed point
5990:{\displaystyle A}
5943:{\displaystyle x}
5923:{\displaystyle A}
5903:{\displaystyle x}
5506:{\displaystyle d}
5386:
4992:
4948:special functions
4787:
4617:
4527:
4456:
4273:
4168:
4079:
4029:
3973:
3900:
3850:
3788:
3524:{\displaystyle n}
3504:{\displaystyle n}
3175:
3131:
3029:
2704:second difference
2360:finite difference
2199:
2166:
2121:
2047:
2014:
1972:
1925:
1895:Pascal's triangle
1868:
1841:
1795:
1762:
1721:
1690:{\displaystyle n}
1670:{\displaystyle k}
1638:
1590:
1550:rational function
1174:linear recurrence
1170:Fibonacci numbers
1164:Fibonacci numbers
985:{\displaystyle n}
898:
751:
490:
390:{\displaystyle n}
362:functions have a
351:{\displaystyle n}
327:{\displaystyle n}
276:{\displaystyle k}
196:Fibonacci numbers
187:{\displaystyle k}
152:{\displaystyle k}
128:{\displaystyle k}
108:{\displaystyle n}
88:{\displaystyle k}
68:{\displaystyle k}
44:{\displaystyle n}
8648:
8613:
8610:"OEIS Index Rec"
8605:
8604:
8586:
8561:
8543:
8521:
8511:
8502:
8500:
8499:
8493:
8482:
8471:
8455:
8443:
8418:
8403:
8364:Ronald L. Rivest
8356:Thomas H. Cormen
8352:
8341:
8324:
8315:
8297:
8296:
8284:
8271:Ljungqvist, Lars
8267:
8261:
8260:
8233:Stokey, Nancy L.
8229:
8223:
8216:
8210:
8203:
8197:
8196:
8190:
8182:
8180:
8179:
8173:
8166:
8158:
8152:
8139:
8133:
8130:
8001:
7999:
7998:
7993:
7991:
7990:
7972:
7971:
7959:
7958:
7940:
7939:
7918:
7917:
7890:
7889:
7872:
7870:
7869:
7864:
7859:
7858:
7834:
7833:
7794:
7793:
7766:
7765:
7746:
7744:
7743:
7738:
7726:
7724:
7723:
7718:
7706:
7704:
7703:
7698:
7696:
7695:
7679:
7677:
7676:
7671:
7659:
7657:
7656:
7651:
7649:
7648:
7629:
7627:
7626:
7621:
7616:
7615:
7594:
7593:
7566:
7565:
7546:
7544:
7543:
7538:
7499:
7497:
7496:
7491:
7474:
7473:
7444:
7442:
7441:
7436:
7434:
7433:
7429:
7407:
7406:
7389:
7387:
7386:
7381:
7373:
7372:
7346:
7344:
7343:
7338:
7323:
7321:
7320:
7315:
7285:Computer science
7266:
7264:
7263:
7258:
7246:
7244:
7243:
7238:
7236:
7235:
7219:
7217:
7216:
7211:
7209:
7208:
7189:
7187:
7186:
7181:
7173:
7172:
7171:
7170:
7141:
7140:
7128:
7127:
7104:
7102:
7101:
7096:
7094:
7093:
7092:
7091:
7071:
7070:
7055:
7054:
6981:
6979:
6978:
6973:
6962:
6961:
6949:
6948:
6933:
6932:
6920:
6919:
6898:
6897:
6885:
6884:
6855:
6853:
6852:
6847:
6840:
6824:
6823:
6805:
6804:
6793:
6790:
6777:
6776:
6758:
6757:
6746:
6743:
6736:
6735:
6717:
6716:
6697:
6695:
6694:
6689:
6678:and a step size
6670:
6668:
6667:
6662:
6657:
6656:
6641:
6640:
6623:
6620:
6577:
6550:When solving an
6529:
6527:
6526:
6521:
6502:
6500:
6499:
6494:
6492:
6491:
6472:
6470:
6469:
6464:
6453:
6445:
6444:
6432:
6424:
6409:
6407:
6406:
6401:
6389:
6387:
6386:
6381:
6366:
6364:
6363:
6358:
6301:
6299:
6298:
6293:
6275:
6273:
6272:
6267:
6245:
6243:
6242:
6237:
6229:
6221:
6220:
6208:
6200:
6182:is smaller than
6181:
6179:
6178:
6173:
6171:
6170:
6154:
6152:
6151:
6146:
6134:
6132:
6131:
6126:
6124:
6123:
6107:
6105:
6104:
6099:
6097:
6096:
6069:
6067:
6066:
6061:
6053:
6052:
6028:
6027:
5996:
5994:
5993:
5988:
5976:
5974:
5973:
5968:
5966:
5965:
5949:
5947:
5946:
5941:
5929:
5927:
5926:
5921:
5909:
5907:
5906:
5901:
5886:
5884:
5883:
5878:
5873:
5872:
5860:
5859:
5832:
5831:
5819:
5818:
5767:
5765:
5764:
5759:
5751:
5750:
5741:
5740:
5722:
5721:
5706:
5705:
5693:
5692:
5677:
5676:
5664:
5663:
5638:
5636:
5635:
5630:
5625:
5624:
5609:
5608:
5590:
5589:
5574:
5573:
5561:
5560:
5545:
5544:
5532:
5531:
5512:
5510:
5509:
5504:
5479:
5477:
5476:
5471:
5469:
5468:
5452:
5450:
5449:
5444:
5442:
5441:
5425:
5423:
5422:
5417:
5415:
5414:
5398:
5396:
5395:
5390:
5388:
5385:
5378:
5377:
5364:
5357:
5356:
5343:
5337:
5336:
5273:
5271:
5270:
5265:
5236:
5235:
5213:
5211:
5210:
5205:
5197:
5196:
5175:
5174:
5138:
5137:
5090:
5088:
5087:
5082:
5068:
5067:
5055:
5054:
5032:
5030:
5029:
5024:
5022:
5021:
5003:
5002:
4993:
4988:
4980:
4975:
4974:
4925:
4923:
4922:
4917:
4899:
4897:
4896:
4891:
4889:
4888:
4870:
4869:
4857:
4856:
4829:
4828:
4803:
4801:
4800:
4795:
4793:
4789:
4788:
4786:
4785:
4784:
4774:
4769:
4753:
4752:
4743:
4740:
4729:
4711:
4710:
4696:
4692:
4691:
4690:
4680:
4669:
4646:
4645:
4628:
4626:
4625:
4620:
4618:
4616:
4615:
4614:
4604:
4599:
4583:
4582:
4573:
4570:
4559:
4541:
4540:
4528:
4526:
4525:
4524:
4514:
4503:
4487:
4486:
4477:
4467:
4465:
4464:
4459:
4457:
4455:
4454:
4453:
4443:
4438:
4422:
4421:
4412:
4409:
4398:
4380:
4379:
4367:
4366:
4351:
4350:
4338:
4337:
4318:
4307:
4284:
4282:
4281:
4276:
4274:
4272:
4271:
4270:
4260:
4255:
4239:
4238:
4229:
4224:
4223:
4211:
4210:
4182:
4180:
4179:
4174:
4169:
4167:
4166:
4165:
4155:
4144:
4128:
4127:
4118:
4113:
4112:
4090:
4088:
4087:
4082:
4080:
4078:
4077:
4076:
4066:
4061:
4045:
4044:
4035:
4030:
4028:
4027:
4026:
4016:
4005:
3989:
3988:
3979:
3974:
3972:
3971:
3970:
3960:
3955:
3939:
3938:
3923:
3911:
3909:
3908:
3903:
3901:
3899:
3898:
3897:
3887:
3882:
3866:
3865:
3856:
3851:
3849:
3848:
3847:
3837:
3832:
3816:
3815:
3814:
3805:
3804:
3794:
3789:
3787:
3786:
3785:
3775:
3770:
3754:
3753:
3738:
3728:
3726:
3725:
3720:
3718:
3717:
3705:
3704:
3695:
3694:
3682:
3681:
3653:
3651:
3650:
3645:
3634:
3633:
3620:
3619:
3607:
3606:
3597:
3596:
3584:
3583:
3530:
3528:
3527:
3522:
3510:
3508:
3507:
3502:
3449:
3447:
3446:
3441:
3436:
3435:
3420:
3419:
3398:
3397:
3366:
3364:
3363:
3358:
3350:
3349:
3334:
3333:
3315:
3314:
3305:
3304:
3275:
3271:
3267:
3263:
3252:
3248:
3240:
3236:
3231:
3230:
3221:
3219:
3218:
3213:
3205:
3204:
3192:
3191:
3182:
3181:
3180:
3167:
3151:
3150:
3138:
3137:
3136:
3123:
3113:
3112:
3100:
3099:
3071:
3069:
3068:
3063:
3058:
3057:
3036:
3035:
3034:
3021:
3014:
3013:
2994:
2989:
2971:
2970:
2961:
2960:
2941:
2939:
2938:
2933:
2928:
2927:
2903:
2902:
2883:
2876:
2874:
2873:
2868:
2863:
2862:
2850:
2849:
2828:
2827:
2809:
2808:
2799:
2798:
2779:
2777:
2776:
2771:
2724:
2723:
2706:
2705:
2697:
2695:
2694:
2689:
2671:
2666:
2665:
2664:first difference
2659:
2657:
2656:
2651:
2646:
2645:
2644:
2628:
2627:
2597:
2595:
2594:
2589:
2584:
2583:
2567:
2565:
2564:
2559:
2547:
2545:
2544:
2539:
2522:in the sequence
2521:
2517:
2515:
2514:
2509:
2507:
2506:
2487:
2485:
2484:
2479:
2464:
2462:
2461:
2456:
2438:
2436:
2435:
2430:
2425:
2424:
2412:
2411:
2393:
2392:
2354:
2352:
2351:
2346:
2270:
2268:
2267:
2262:
2234:
2233:
2219:
2217:
2216:
2211:
2206:
2205:
2204:
2198:
2183:
2173:
2172:
2171:
2158:
2144:
2142:
2141:
2136:
2128:
2127:
2126:
2113:
2096:
2094:
2093:
2088:
2080:
2054:
2053:
2052:
2046:
2031:
2021:
2020:
2019:
2006:
1986:
1984:
1983:
1978:
1973:
1971:
1945:
1937:
1932:
1931:
1930:
1917:
1892:
1890:
1889:
1884:
1876:
1874:
1873:
1860:
1849:
1847:
1846:
1833:
1815:
1813:
1812:
1807:
1802:
1801:
1800:
1791:
1779:
1769:
1768:
1767:
1761:
1750:
1738:
1728:
1727:
1726:
1713:
1696:
1694:
1693:
1688:
1676:
1674:
1673:
1668:
1656:
1654:
1653:
1648:
1646:
1644:
1643:
1630:
1604:
1602:
1601:
1596:
1591:
1589:
1588:
1587:
1562:
1543:
1541:
1540:
1535:
1521:
1520:
1486:
1484:
1483:
1478:
1476:
1475:
1463:
1462:
1450:
1449:
1432:
1430:
1429:
1424:
1422:
1421:
1409:
1408:
1396:
1395:
1378:
1376:
1375:
1370:
1368:
1367:
1355:
1354:
1342:
1341:
1319:
1317:
1316:
1311:
1303:
1302:
1285:
1283:
1282:
1277:
1269:
1268:
1243:
1241:
1240:
1235:
1233:
1232:
1214:
1213:
1195:
1194:
1159:
1157:
1156:
1151:
1149:
1148:
1132:
1130:
1129:
1124:
1109:
1107:
1106:
1101:
1083:
1081:
1080:
1075:
1067:
1066:
1048:
1047:
1032:
1031:
991:
989:
988:
983:
968:
957:
955:
954:
949:
922:
920:
919:
914:
899:
896:
833:
829:
825:
823:
822:
817:
809:
808:
796:
775:
773:
772:
767:
752:
749:
743:
742:
718:
717:
699:
698:
668:
667:
648:
627:
625:
624:
619:
617:
616:
600:
598:
597:
592:
584:
583:
567:
560:
558:
557:
552:
538:
514:
512:
511:
506:
491:
488:
482:
481:
451:
450:
406:indexed families
396:
394:
393:
388:
357:
355:
354:
349:
333:
331:
330:
325:
309:
307:
306:
301:
282:
280:
279:
274:
263:where the order
262:
260:
259:
254:
252:
251:
233:
232:
214:
213:
193:
191:
190:
185:
169:
158:
156:
155:
150:
134:
132:
131:
126:
114:
112:
111:
106:
94:
92:
91:
86:
74:
72:
71:
66:
50:
48:
47:
42:
8658:
8657:
8651:
8650:
8649:
8647:
8646:
8645:
8621:
8620:
8608:
8571:
8568:
8497:
8495:
8491:
8480:
8468:
8440:
8400:
8306:
8301:
8300:
8293:
8268:
8264:
8257:
8230:
8226:
8217:
8213:
8204:
8200:
8184:
8183:
8177:
8175:
8171:
8164:
8162:"Archived copy"
8160:
8159:
8155:
8140:
8136:
8131:
8127:
8122:
8117:
8112:
8038:
8021:
8011:
7977:
7973:
7967:
7963:
7948:
7944:
7935:
7931:
7913:
7909:
7885:
7881:
7879:
7876:
7875:
7845:
7841:
7823:
7819:
7789:
7785:
7761:
7757:
7755:
7752:
7751:
7732:
7729:
7728:
7712:
7709:
7708:
7691:
7687:
7685:
7682:
7681:
7665:
7662:
7661:
7644:
7640:
7638:
7635:
7634:
7605:
7601:
7589:
7585:
7561:
7557:
7555:
7552:
7551:
7532:
7529:
7528:
7518:digital filters
7506:
7469:
7465:
7457:
7454:
7453:
7450:time complexity
7425:
7421:
7417:
7402:
7398:
7396:
7393:
7392:
7368:
7364:
7362:
7359:
7358:
7332:
7329:
7328:
7309:
7306:
7305:
7287:
7252:
7249:
7248:
7231:
7227:
7225:
7222:
7221:
7204:
7200:
7198:
7195:
7194:
7166:
7162:
7155:
7151:
7136:
7132:
7117:
7113:
7111:
7108:
7107:
7087:
7083:
7076:
7072:
7066:
7062:
7044:
7040:
7038:
7035:
7034:
7000:
6995:
6957:
6953:
6944:
6940:
6928:
6924:
6915:
6911:
6893:
6889:
6874:
6870:
6867:
6864:
6863:
6819:
6815:
6800:
6796:
6772:
6768:
6753:
6749:
6731:
6727:
6712:
6708:
6706:
6703:
6702:
6683:
6680:
6679:
6652:
6648:
6636:
6632:
6570:
6568:
6565:
6564:
6548:
6515:
6512:
6511:
6487:
6483:
6481:
6478:
6477:
6449:
6440:
6436:
6425:
6420:
6418:
6415:
6414:
6395:
6392:
6391:
6375:
6372:
6371:
6310:
6307:
6306:
6281:
6278:
6277:
6261:
6258:
6257:
6225:
6216:
6212:
6201:
6196:
6194:
6191:
6190:
6166:
6162:
6160:
6157:
6156:
6140:
6137:
6136:
6119:
6115:
6113:
6110:
6109:
6092:
6088:
6086:
6083:
6082:
6042:
6038:
6023:
6019:
6017:
6014:
6013:
6007:
5982:
5979:
5978:
5961:
5957:
5955:
5952:
5951:
5935:
5932:
5931:
5915:
5912:
5911:
5895:
5892:
5891:
5868:
5864:
5849:
5845:
5827:
5823:
5814:
5810:
5805:
5802:
5801:
5795:
5789:
5746:
5742:
5736:
5732:
5711:
5707:
5701:
5697:
5682:
5678:
5672:
5668:
5659:
5655:
5653:
5650:
5649:
5614:
5610:
5604:
5600:
5579:
5575:
5569:
5565:
5550:
5546:
5540:
5536:
5527:
5523:
5521:
5518:
5517:
5498:
5495:
5494:
5491:
5486:
5464:
5460:
5458:
5455:
5454:
5437:
5433:
5431:
5428:
5427:
5410:
5406:
5404:
5401:
5400:
5373:
5369:
5365:
5352:
5348:
5344:
5341:
5326:
5322:
5320:
5317:
5316:
5313:
5307:
5231:
5227:
5225:
5222:
5221:
5186:
5182:
5170:
5166:
5127:
5123:
5106:
5103:
5102:
5096:Bessel function
5063:
5059:
5050:
5046:
5044:
5041:
5040:
5011:
5007:
4998:
4994:
4981:
4979:
4964:
4960:
4958:
4955:
4954:
4936:
4905:
4902:
4901:
4881:
4877:
4865:
4861:
4849:
4845:
4818:
4814:
4812:
4809:
4808:
4780:
4776:
4770:
4759:
4754:
4748:
4744:
4742:
4730:
4719:
4706:
4702:
4701:
4697:
4686:
4682:
4670:
4659:
4654:
4650:
4641:
4637:
4635:
4632:
4631:
4610:
4606:
4600:
4589:
4584:
4578:
4574:
4572:
4560:
4549:
4536:
4532:
4520:
4516:
4504:
4493:
4488:
4482:
4478:
4476:
4474:
4471:
4470:
4449:
4445:
4439:
4428:
4423:
4417:
4413:
4411:
4399:
4388:
4375:
4371:
4362:
4358:
4346:
4342:
4327:
4323:
4308:
4297:
4291:
4288:
4287:
4266:
4262:
4256:
4245:
4240:
4234:
4230:
4228:
4219:
4215:
4200:
4196:
4194:
4191:
4190:
4161:
4157:
4145:
4134:
4129:
4123:
4119:
4117:
4108:
4104:
4102:
4099:
4098:
4072:
4068:
4062:
4051:
4046:
4040:
4036:
4034:
4022:
4018:
4006:
3995:
3990:
3984:
3980:
3978:
3966:
3962:
3956:
3945:
3940:
3928:
3924:
3922:
3920:
3917:
3916:
3893:
3889:
3883:
3872:
3867:
3861:
3857:
3855:
3843:
3839:
3833:
3822:
3817:
3810:
3806:
3800:
3796:
3795:
3793:
3781:
3777:
3771:
3760:
3755:
3743:
3739:
3737:
3735:
3732:
3731:
3713:
3709:
3700:
3696:
3690:
3686:
3671:
3667:
3665:
3662:
3661:
3629:
3625:
3615:
3611:
3602:
3598:
3592:
3588:
3573:
3569:
3567:
3564:
3563:
3557:
3552:
3546:
3541:
3516:
3513:
3512:
3496:
3493:
3492:
3489:
3431:
3427:
3409:
3405:
3387:
3383:
3378:
3375:
3374:
3345:
3341:
3329:
3325:
3310:
3306:
3300:
3296:
3291:
3288:
3287:
3273:
3269:
3265:
3261:
3257:of a function.
3250:
3246:
3238:
3234:
3228:
3227:
3200:
3196:
3187:
3183:
3176:
3163:
3162:
3161:
3146:
3142:
3132:
3119:
3118:
3117:
3108:
3104:
3089:
3085:
3083:
3080:
3079:
3041:
3037:
3030:
3017:
3016:
3015:
3009:
3005:
2990:
2979:
2966:
2962:
2956:
2952:
2950:
2947:
2946:
2917:
2913:
2898:
2894:
2892:
2889:
2888:
2881:
2858:
2854:
2839:
2835:
2817:
2813:
2804:
2800:
2794:
2790:
2788:
2785:
2784:
2719:
2715:
2713:
2710:
2709:
2703:
2702:
2677:
2674:
2673:
2669:
2663:
2662:
2640:
2633:
2629:
2623:
2619:
2608:
2605:
2604:
2579:
2575:
2573:
2570:
2569:
2553:
2550:
2549:
2527:
2524:
2523:
2519:
2502:
2498:
2493:
2490:
2489:
2470:
2467:
2466:
2447:
2444:
2443:
2420:
2416:
2401:
2397:
2388:
2384:
2373:
2370:
2369:
2283:
2280:
2279:
2253:
2250:
2249:
2231:
2230:
2226:
2200:
2188:
2179:
2178:
2177:
2167:
2154:
2153:
2152:
2150:
2147:
2146:
2122:
2109:
2108:
2107:
2105:
2102:
2101:
2076:
2048:
2036:
2027:
2026:
2025:
2015:
2002:
2001:
2000:
1998:
1995:
1994:
1946:
1938:
1936:
1926:
1913:
1912:
1911:
1909:
1906:
1905:
1869:
1856:
1855:
1853:
1842:
1829:
1828:
1826:
1824:
1821:
1820:
1796:
1781:
1775:
1774:
1773:
1763:
1751:
1740:
1734:
1733:
1732:
1722:
1709:
1708:
1707:
1705:
1702:
1701:
1682:
1679:
1678:
1662:
1659:
1658:
1639:
1626:
1625:
1623:
1621:
1618:
1617:
1611:
1583:
1579:
1566:
1561:
1559:
1556:
1555:
1516:
1512:
1510:
1507:
1506:
1503:Binet's formula
1471:
1467:
1458:
1454:
1445:
1441:
1439:
1436:
1435:
1417:
1413:
1404:
1400:
1391:
1387:
1385:
1382:
1381:
1363:
1359:
1350:
1346:
1337:
1333:
1331:
1328:
1327:
1298:
1294:
1292:
1289:
1288:
1264:
1260:
1258:
1255:
1254:
1222:
1218:
1203:
1199:
1190:
1186:
1184:
1181:
1180:
1166:
1144:
1140:
1138:
1135:
1134:
1115:
1112:
1111:
1092:
1089:
1088:
1062:
1058:
1043:
1039:
1021:
1017:
1015:
1012:
1011:
1001:
977:
974:
973:
966:
934:
931:
930:
895:
859:
856:
855:
845:
840:
831:
827:
804:
800:
792:
784:
781:
780:
748:
732:
728:
707:
703:
688:
684:
663:
659:
657:
654:
653:
646:
612:
608:
606:
603:
602:
579:
575:
573:
570:
569:
565:
534:
526:
523:
522:
487:
471:
467:
446:
442:
440:
437:
436:
422:
414:natural numbers
382:
379:
378:
343:
340:
339:
319:
316:
315:
292:
289:
288:
268:
265:
264:
241:
237:
222:
218:
209:
205:
203:
200:
199:
179:
176:
175:
172:linear function
167:
144:
141:
140:
120:
117:
116:
100:
97:
96:
80:
77:
76:
60:
57:
56:
36:
33:
32:
17:
12:
11:
5:
8656:
8655:
8644:
8643:
8638:
8633:
8619:
8618:
8606:
8587:
8567:
8566:External links
8564:
8563:
8562:
8534:(2): 215–235.
8523:
8513:
8503:
8473:
8466:
8445:
8438:
8419:
8417:on 2014-11-10.
8404:
8398:
8385:
8368:Clifford Stein
8353:
8342:
8325:
8316:
8305:
8302:
8299:
8298:
8291:
8262:
8255:
8224:
8211:
8198:
8153:
8134:
8124:
8123:
8121:
8118:
8116:
8113:
8111:
8110:
8105:
8100:
8095:
8090:
8085:
8080:
8075:
8070:
8065:
8060:
8055:
8050:
8045:
8039:
8037:
8034:
8010:
8007:
8003:
8002:
7989:
7986:
7983:
7980:
7976:
7970:
7966:
7962:
7957:
7954:
7951:
7947:
7943:
7938:
7934:
7930:
7927:
7924:
7921:
7916:
7912:
7908:
7905:
7902:
7899:
7896:
7893:
7888:
7884:
7873:
7862:
7857:
7854:
7851:
7848:
7844:
7840:
7837:
7832:
7829:
7826:
7822:
7818:
7815:
7812:
7809:
7806:
7803:
7800:
7797:
7792:
7788:
7784:
7781:
7778:
7775:
7772:
7769:
7764:
7760:
7736:
7716:
7694:
7690:
7669:
7647:
7643:
7631:
7630:
7619:
7614:
7611:
7608:
7604:
7600:
7597:
7592:
7588:
7584:
7581:
7578:
7575:
7572:
7569:
7564:
7560:
7536:
7505:
7502:
7489:
7486:
7483:
7480:
7477:
7472:
7468:
7464:
7461:
7446:
7445:
7432:
7428:
7424:
7420:
7416:
7413:
7410:
7405:
7401:
7390:
7379:
7376:
7371:
7367:
7336:
7313:
7286:
7283:
7256:
7234:
7230:
7207:
7203:
7191:
7190:
7179:
7176:
7169:
7165:
7161:
7158:
7154:
7150:
7147:
7144:
7139:
7135:
7131:
7126:
7123:
7120:
7116:
7105:
7090:
7086:
7082:
7079:
7075:
7069:
7065:
7061:
7058:
7053:
7050:
7047:
7043:
6999:
6996:
6994:
6991:
6987:discretization
6983:
6982:
6971:
6968:
6965:
6960:
6956:
6952:
6947:
6943:
6939:
6936:
6931:
6927:
6923:
6918:
6914:
6910:
6907:
6904:
6901:
6896:
6892:
6888:
6883:
6880:
6877:
6873:
6857:
6856:
6845:
6839:
6836:
6833:
6830:
6827:
6822:
6818:
6814:
6811:
6808:
6803:
6799:
6789:
6786:
6783:
6780:
6775:
6771:
6767:
6764:
6761:
6756:
6752:
6742:
6739:
6734:
6730:
6726:
6723:
6720:
6715:
6711:
6687:
6676:Euler's method
6672:
6671:
6660:
6655:
6651:
6647:
6644:
6639:
6635:
6631:
6628:
6619:
6616:
6613:
6610:
6607:
6604:
6601:
6598:
6595:
6592:
6589:
6586:
6583:
6580:
6576:
6573:
6547:
6544:
6519:
6490:
6486:
6474:
6473:
6462:
6459:
6456:
6452:
6448:
6443:
6439:
6435:
6431:
6428:
6423:
6399:
6379:
6368:
6367:
6356:
6353:
6350:
6347:
6344:
6341:
6338:
6335:
6332:
6329:
6326:
6323:
6320:
6317:
6314:
6291:
6288:
6285:
6265:
6247:
6246:
6235:
6232:
6228:
6224:
6219:
6215:
6211:
6207:
6204:
6199:
6169:
6165:
6144:
6122:
6118:
6095:
6091:
6075:locally stable
6071:
6070:
6059:
6056:
6051:
6048:
6045:
6041:
6037:
6034:
6031:
6026:
6022:
6006:
6003:
5999:absolute value
5986:
5964:
5960:
5939:
5919:
5899:
5888:
5887:
5876:
5871:
5867:
5863:
5858:
5855:
5852:
5848:
5844:
5841:
5838:
5835:
5830:
5826:
5822:
5817:
5813:
5809:
5791:Main article:
5788:
5785:
5769:
5768:
5757:
5754:
5749:
5745:
5739:
5735:
5731:
5728:
5725:
5720:
5717:
5714:
5710:
5704:
5700:
5696:
5691:
5688:
5685:
5681:
5675:
5671:
5667:
5662:
5658:
5640:
5639:
5628:
5623:
5620:
5617:
5613:
5607:
5603:
5599:
5596:
5593:
5588:
5585:
5582:
5578:
5572:
5568:
5564:
5559:
5556:
5553:
5549:
5543:
5539:
5535:
5530:
5526:
5502:
5490:
5487:
5485:
5482:
5467:
5463:
5440:
5436:
5413:
5409:
5384:
5381:
5376:
5372:
5368:
5363:
5360:
5355:
5351:
5347:
5340:
5335:
5332:
5329:
5325:
5309:Main article:
5306:
5303:
5299:hypergeometric
5275:
5274:
5263:
5260:
5257:
5254:
5251:
5248:
5245:
5242:
5239:
5234:
5230:
5215:
5214:
5203:
5200:
5195:
5192:
5189:
5185:
5181:
5178:
5173:
5169:
5165:
5162:
5159:
5156:
5153:
5150:
5147:
5144:
5141:
5136:
5133:
5130:
5126:
5122:
5119:
5116:
5113:
5110:
5092:
5091:
5080:
5077:
5074:
5071:
5066:
5062:
5058:
5053:
5049:
5034:
5033:
5020:
5017:
5014:
5010:
5006:
5001:
4997:
4991:
4987:
4984:
4978:
4973:
4970:
4967:
4963:
4935:
4932:
4915:
4912:
4909:
4887:
4884:
4880:
4876:
4873:
4868:
4864:
4860:
4855:
4852:
4848:
4844:
4841:
4838:
4835:
4832:
4827:
4824:
4821:
4817:
4805:
4804:
4792:
4783:
4779:
4773:
4768:
4765:
4762:
4758:
4751:
4747:
4739:
4736:
4733:
4728:
4725:
4722:
4718:
4714:
4709:
4705:
4700:
4695:
4689:
4685:
4679:
4676:
4673:
4668:
4665:
4662:
4658:
4653:
4649:
4644:
4640:
4629:
4613:
4609:
4603:
4598:
4595:
4592:
4588:
4581:
4577:
4569:
4566:
4563:
4558:
4555:
4552:
4548:
4544:
4539:
4535:
4531:
4523:
4519:
4513:
4510:
4507:
4502:
4499:
4496:
4492:
4485:
4481:
4468:
4452:
4448:
4442:
4437:
4434:
4431:
4427:
4420:
4416:
4408:
4405:
4402:
4397:
4394:
4391:
4387:
4383:
4378:
4374:
4370:
4365:
4361:
4357:
4354:
4349:
4345:
4341:
4336:
4333:
4330:
4326:
4322:
4317:
4314:
4311:
4306:
4303:
4300:
4296:
4285:
4269:
4265:
4259:
4254:
4251:
4248:
4244:
4237:
4233:
4227:
4222:
4218:
4214:
4209:
4206:
4203:
4199:
4184:
4183:
4172:
4164:
4160:
4154:
4151:
4148:
4143:
4140:
4137:
4133:
4126:
4122:
4116:
4111:
4107:
4092:
4091:
4075:
4071:
4065:
4060:
4057:
4054:
4050:
4043:
4039:
4033:
4025:
4021:
4015:
4012:
4009:
4004:
4001:
3998:
3994:
3987:
3983:
3977:
3969:
3965:
3959:
3954:
3951:
3948:
3944:
3937:
3934:
3931:
3927:
3913:
3912:
3896:
3892:
3886:
3881:
3878:
3875:
3871:
3864:
3860:
3854:
3846:
3842:
3836:
3831:
3828:
3825:
3821:
3813:
3809:
3803:
3799:
3792:
3784:
3780:
3774:
3769:
3766:
3763:
3759:
3752:
3749:
3746:
3742:
3729:
3716:
3712:
3708:
3703:
3699:
3693:
3689:
3685:
3680:
3677:
3674:
3670:
3655:
3654:
3643:
3640:
3637:
3632:
3628:
3623:
3618:
3614:
3610:
3605:
3601:
3595:
3591:
3587:
3582:
3579:
3576:
3572:
3556:
3553:
3548:Main article:
3545:
3542:
3540:
3537:
3520:
3500:
3488:
3485:
3451:
3450:
3439:
3434:
3430:
3426:
3423:
3418:
3415:
3412:
3408:
3404:
3401:
3396:
3393:
3390:
3386:
3382:
3368:
3367:
3356:
3353:
3348:
3344:
3340:
3337:
3332:
3328:
3324:
3321:
3318:
3313:
3309:
3303:
3299:
3295:
3223:
3222:
3211:
3208:
3203:
3199:
3195:
3190:
3186:
3179:
3174:
3171:
3166:
3160:
3157:
3154:
3149:
3145:
3141:
3135:
3130:
3127:
3122:
3116:
3111:
3107:
3103:
3098:
3095:
3092:
3088:
3073:
3072:
3061:
3056:
3053:
3050:
3047:
3044:
3040:
3033:
3028:
3025:
3020:
3012:
3008:
3004:
3001:
2998:
2993:
2988:
2985:
2982:
2978:
2974:
2969:
2965:
2959:
2955:
2931:
2926:
2923:
2920:
2916:
2912:
2909:
2906:
2901:
2897:
2878:
2877:
2866:
2861:
2857:
2853:
2848:
2845:
2842:
2838:
2834:
2831:
2826:
2823:
2820:
2816:
2812:
2807:
2803:
2797:
2793:
2769:
2766:
2763:
2760:
2757:
2754:
2751:
2748:
2745:
2742:
2739:
2736:
2733:
2730:
2727:
2722:
2718:
2687:
2684:
2681:
2649:
2643:
2639:
2636:
2632:
2626:
2622:
2618:
2615:
2612:
2587:
2582:
2578:
2557:
2537:
2534:
2531:
2505:
2501:
2497:
2477:
2474:
2454:
2451:
2440:
2439:
2428:
2423:
2419:
2415:
2410:
2407:
2404:
2400:
2396:
2391:
2387:
2383:
2380:
2377:
2356:
2355:
2344:
2341:
2338:
2335:
2332:
2329:
2326:
2323:
2320:
2317:
2314:
2311:
2308:
2305:
2302:
2299:
2296:
2293:
2290:
2287:
2260:
2257:
2225:
2222:
2209:
2203:
2197:
2194:
2191:
2187:
2182:
2176:
2170:
2165:
2162:
2157:
2134:
2131:
2125:
2120:
2117:
2112:
2098:
2097:
2086:
2083:
2079:
2075:
2072:
2069:
2066:
2063:
2060:
2057:
2051:
2045:
2042:
2039:
2035:
2030:
2024:
2018:
2013:
2010:
2005:
1988:
1987:
1976:
1970:
1967:
1964:
1961:
1958:
1955:
1952:
1949:
1944:
1941:
1935:
1929:
1924:
1921:
1916:
1882:
1879:
1872:
1867:
1864:
1859:
1852:
1845:
1840:
1837:
1832:
1817:
1816:
1805:
1799:
1794:
1790:
1787:
1784:
1778:
1772:
1766:
1760:
1757:
1754:
1749:
1746:
1743:
1737:
1731:
1725:
1720:
1717:
1712:
1686:
1666:
1642:
1637:
1634:
1629:
1610:
1607:
1606:
1605:
1594:
1586:
1582:
1578:
1575:
1572:
1569:
1565:
1533:
1530:
1527:
1524:
1519:
1515:
1499:
1498:
1488:
1487:
1474:
1470:
1466:
1461:
1457:
1453:
1448:
1444:
1433:
1420:
1416:
1412:
1407:
1403:
1399:
1394:
1390:
1379:
1366:
1362:
1358:
1353:
1349:
1345:
1340:
1336:
1321:
1320:
1309:
1306:
1301:
1297:
1286:
1275:
1272:
1267:
1263:
1245:
1244:
1231:
1228:
1225:
1221:
1217:
1212:
1209:
1206:
1202:
1198:
1193:
1189:
1165:
1162:
1147:
1143:
1122:
1119:
1099:
1096:
1085:
1084:
1073:
1070:
1065:
1061:
1057:
1054:
1051:
1046:
1042:
1038:
1035:
1030:
1027:
1024:
1020:
1000:
997:
993:
992:
981:
959:
958:
947:
944:
941:
938:
924:
923:
912:
909:
906:
903:
893:
890:
887:
884:
881:
878:
875:
872:
869:
866:
863:
844:
841:
839:
836:
815:
812:
807:
803:
799:
795:
791:
788:
777:
776:
765:
762:
759:
756:
746:
741:
738:
735:
731:
727:
724:
721:
716:
713:
710:
706:
702:
697:
694:
691:
687:
683:
680:
677:
674:
671:
666:
662:
649:has the form
615:
611:
590:
587:
582:
578:
562:
561:
550:
547:
544:
541:
537:
533:
530:
516:
515:
504:
501:
498:
495:
485:
480:
477:
474:
470:
466:
463:
460:
457:
454:
449:
445:
421:
418:
386:
347:
323:
299:
296:
272:
250:
247:
244:
240:
236:
231:
228:
225:
221:
217:
212:
208:
183:
148:
135:is called the
124:
115:; this number
104:
84:
64:
40:
15:
9:
6:
4:
3:
2:
8654:
8653:
8642:
8641:Combinatorics
8639:
8637:
8634:
8632:
8629:
8628:
8626:
8616:
8611:
8607:
8602:
8601:
8596:
8593:
8588:
8584:
8580:
8579:
8574:
8570:
8569:
8559:
8555:
8551:
8547:
8542:
8537:
8533:
8529:
8524:
8519:
8514:
8509:
8504:
8494:on 2016-03-04
8490:
8486:
8479:
8474:
8469:
8467:0-273-70195-9
8463:
8459:
8454:
8453:
8446:
8441:
8439:0-387-23234-6
8435:
8431:
8430:
8425:
8424:Flahive, Mary
8420:
8416:
8412:
8411:
8405:
8401:
8399:0-201-55802-5
8395:
8391:
8386:
8383:
8382:0-262-03293-7
8379:
8375:
8374:
8369:
8365:
8361:
8357:
8354:
8350:
8349:
8343:
8339:
8335:
8331:
8326:
8322:
8317:
8313:
8308:
8307:
8294:
8292:0-262-12274-X
8288:
8283:
8282:
8276:
8272:
8266:
8258:
8256:0-674-75096-9
8252:
8248:
8247:
8242:
8238:
8234:
8228:
8221:
8215:
8208:
8202:
8194:
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8170:
8163:
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8147:
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8138:
8129:
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8104:
8101:
8099:
8096:
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8059:
8056:
8054:
8051:
8049:
8046:
8044:
8041:
8040:
8033:
8031:
8027:
8026:interest rate
8020:
8016:
8006:
7987:
7984:
7981:
7978:
7974:
7968:
7964:
7960:
7955:
7952:
7949:
7945:
7936:
7932:
7928:
7925:
7919:
7914:
7910:
7903:
7900:
7897:
7891:
7886:
7882:
7874:
7855:
7852:
7849:
7846:
7842:
7838:
7835:
7830:
7827:
7824:
7820:
7813:
7810:
7807:
7798:
7795:
7790:
7786:
7779:
7776:
7773:
7767:
7762:
7758:
7750:
7749:
7748:
7734:
7714:
7692:
7688:
7667:
7645:
7641:
7617:
7612:
7609:
7606:
7602:
7598:
7595:
7590:
7586:
7579:
7576:
7573:
7567:
7562:
7558:
7550:
7549:
7548:
7534:
7526:
7521:
7519:
7515:
7511:
7501:
7481:
7475:
7470:
7466:
7459:
7451:
7430:
7426:
7422:
7418:
7414:
7411:
7408:
7403:
7399:
7391:
7377:
7374:
7369:
7365:
7357:
7356:
7355:
7353:
7352:binary search
7348:
7334:
7325:
7311:
7302:
7300:
7296:
7292:
7282:
7281:populations.
7280:
7276:
7272:
7268:
7254:
7232:
7228:
7205:
7201:
7177:
7167:
7163:
7159:
7156:
7152:
7148:
7145:
7137:
7133:
7129:
7124:
7121:
7118:
7114:
7106:
7088:
7084:
7080:
7077:
7073:
7067:
7063:
7059:
7056:
7051:
7048:
7045:
7041:
7033:
7032:
7031:
7029:
7025:
7021:
7016:
7011:
7009:
7005:
6990:
6988:
6969:
6966:
6963:
6958:
6954:
6950:
6945:
6941:
6937:
6929:
6925:
6921:
6916:
6912:
6905:
6902:
6899:
6894:
6890:
6886:
6881:
6878:
6875:
6871:
6862:
6861:
6860:
6843:
6837:
6831:
6828:
6825:
6820:
6816:
6809:
6806:
6801:
6797:
6787:
6781:
6778:
6773:
6769:
6762:
6759:
6754:
6750:
6740:
6732:
6728:
6721:
6718:
6713:
6709:
6701:
6700:
6699:
6685:
6677:
6658:
6653:
6649:
6645:
6637:
6633:
6626:
6617:
6608:
6602:
6599:
6596:
6590:
6587:
6581:
6574:
6571:
6563:
6562:
6561:
6560:
6556:
6553:
6543:
6541:
6537:
6533:
6517:
6509:
6504:
6488:
6484:
6460:
6457:
6454:
6441:
6437:
6429:
6426:
6413:
6412:
6411:
6397:
6377:
6351:
6345:
6342:
6339:
6336:
6333:
6330:
6327:
6324:
6318:
6312:
6305:
6304:
6303:
6289:
6286:
6283:
6263:
6254:
6252:
6233:
6230:
6217:
6213:
6205:
6202:
6189:
6188:
6187:
6185:
6167:
6163:
6142:
6120:
6116:
6093:
6089:
6080:
6076:
6057:
6049:
6046:
6043:
6039:
6032:
6029:
6024:
6020:
6012:
6011:
6010:
6002:
6000:
5984:
5962:
5958:
5937:
5917:
5897:
5869:
5865:
5861:
5856:
5853:
5850:
5846:
5839:
5836:
5828:
5824:
5820:
5815:
5811:
5800:
5799:
5798:
5794:
5784:
5782:
5778:
5774:
5755:
5752:
5747:
5743:
5737:
5733:
5729:
5726:
5723:
5718:
5715:
5712:
5708:
5702:
5698:
5694:
5689:
5686:
5683:
5679:
5673:
5669:
5665:
5660:
5656:
5648:
5647:
5646:
5645:
5626:
5621:
5618:
5615:
5611:
5605:
5601:
5597:
5594:
5591:
5586:
5583:
5580:
5576:
5570:
5566:
5562:
5557:
5554:
5551:
5547:
5541:
5537:
5533:
5528:
5524:
5516:
5515:
5514:
5500:
5481:
5465:
5461:
5438:
5434:
5411:
5407:
5382:
5379:
5374:
5370:
5366:
5361:
5358:
5353:
5349:
5345:
5338:
5333:
5330:
5327:
5323:
5312:
5302:
5300:
5296:
5292:
5288:
5284:
5280:
5258:
5255:
5252:
5249:
5246:
5240:
5237:
5232:
5228:
5220:
5219:
5218:
5217:is solved by
5201:
5198:
5193:
5190:
5187:
5183:
5179:
5176:
5171:
5167:
5160:
5157:
5154:
5151:
5148:
5145:
5139:
5134:
5131:
5128:
5124:
5117:
5114:
5111:
5101:
5100:
5099:
5097:
5078:
5072:
5064:
5060:
5056:
5051:
5047:
5039:
5038:
5037:
5018:
5015:
5012:
5008:
5004:
4999:
4995:
4989:
4985:
4982:
4976:
4971:
4968:
4965:
4961:
4953:
4952:
4951:
4949:
4945:
4941:
4931:
4929:
4913:
4907:
4885:
4882:
4878:
4874:
4871:
4866:
4862:
4853:
4850:
4846:
4842:
4839:
4836:
4830:
4825:
4822:
4819:
4815:
4790:
4781:
4777:
4771:
4766:
4763:
4760:
4756:
4749:
4745:
4737:
4734:
4731:
4726:
4723:
4720:
4716:
4712:
4707:
4703:
4698:
4693:
4687:
4683:
4677:
4674:
4671:
4666:
4663:
4660:
4656:
4651:
4647:
4642:
4638:
4630:
4611:
4607:
4601:
4596:
4593:
4590:
4586:
4579:
4575:
4567:
4564:
4561:
4556:
4553:
4550:
4546:
4542:
4537:
4533:
4529:
4521:
4517:
4511:
4508:
4505:
4500:
4497:
4494:
4490:
4483:
4479:
4469:
4450:
4446:
4440:
4435:
4432:
4429:
4425:
4418:
4414:
4406:
4403:
4400:
4395:
4392:
4389:
4385:
4381:
4376:
4372:
4368:
4363:
4359:
4355:
4347:
4343:
4339:
4334:
4331:
4328:
4324:
4315:
4312:
4309:
4304:
4301:
4298:
4294:
4286:
4267:
4263:
4257:
4252:
4249:
4246:
4242:
4235:
4231:
4225:
4220:
4216:
4212:
4207:
4204:
4201:
4197:
4189:
4188:
4187:
4170:
4162:
4158:
4152:
4149:
4146:
4141:
4138:
4135:
4131:
4124:
4120:
4114:
4109:
4105:
4097:
4096:
4095:
4073:
4069:
4063:
4058:
4055:
4052:
4048:
4041:
4037:
4031:
4023:
4019:
4013:
4010:
4007:
4002:
3999:
3996:
3992:
3985:
3981:
3975:
3967:
3963:
3957:
3952:
3949:
3946:
3942:
3935:
3932:
3929:
3925:
3915:
3914:
3894:
3890:
3884:
3879:
3876:
3873:
3869:
3862:
3858:
3852:
3844:
3840:
3834:
3829:
3826:
3823:
3819:
3811:
3807:
3801:
3797:
3790:
3782:
3778:
3772:
3767:
3764:
3761:
3757:
3750:
3747:
3744:
3740:
3730:
3714:
3710:
3706:
3701:
3697:
3691:
3687:
3683:
3678:
3675:
3672:
3668:
3660:
3659:
3658:
3641:
3638:
3635:
3630:
3626:
3621:
3616:
3612:
3608:
3603:
3599:
3593:
3589:
3585:
3580:
3577:
3574:
3570:
3562:
3561:
3560:
3551:
3536:
3534:
3518:
3498:
3484:
3482:
3478:
3474:
3470:
3466:
3464:
3460:
3454:
3437:
3432:
3428:
3424:
3421:
3416:
3413:
3410:
3406:
3402:
3399:
3394:
3391:
3388:
3384:
3380:
3373:
3372:
3371:
3354:
3351:
3346:
3342:
3338:
3335:
3330:
3326:
3319:
3316:
3311:
3307:
3301:
3293:
3286:
3285:
3284:
3281:
3279:
3258:
3256:
3244:
3232:
3209:
3201:
3197:
3188:
3172:
3169:
3158:
3155:
3152:
3147:
3143:
3128:
3125:
3114:
3109:
3105:
3101:
3096:
3093:
3090:
3086:
3078:
3077:
3076:
3059:
3054:
3051:
3048:
3045:
3042:
3038:
3026:
3023:
3010:
3002:
2999:
2991:
2986:
2983:
2980:
2976:
2972:
2967:
2963:
2957:
2945:
2944:
2943:
2929:
2924:
2921:
2918:
2910:
2904:
2899:
2886:
2885:th difference
2864:
2859:
2855:
2851:
2846:
2843:
2840:
2836:
2832:
2829:
2824:
2821:
2818:
2814:
2810:
2805:
2801:
2795:
2783:
2782:
2781:
2767:
2761:
2749:
2746:
2737:
2728:
2725:
2720:
2707:
2698:
2685:
2682:
2667:
2647:
2637:
2634:
2624:
2620:
2613:
2610:
2603:
2598:
2585:
2580:
2576:
2535:
2532:
2503:
2499:
2475:
2452:
2426:
2421:
2417:
2413:
2408:
2405:
2402:
2398:
2394:
2389:
2381:
2368:
2367:
2366:
2363:
2361:
2342:
2336:
2330:
2327:
2321:
2318:
2315:
2309:
2306:
2300:
2291:
2278:
2277:
2276:
2274:
2258:
2247:
2243:
2239:
2235:
2221:
2207:
2195:
2192:
2189:
2185:
2174:
2163:
2160:
2132:
2129:
2118:
2115:
2084:
2081:
2077:
2070:
2067:
2064:
2061:
2058:
2043:
2040:
2037:
2033:
2022:
2011:
2008:
1993:
1992:
1991:
1974:
1968:
1962:
1959:
1956:
1950:
1947:
1942:
1939:
1933:
1922:
1919:
1904:
1903:
1902:
1900:
1896:
1880:
1877:
1865:
1862:
1850:
1838:
1835:
1803:
1792:
1788:
1785:
1782:
1770:
1758:
1755:
1752:
1747:
1744:
1741:
1729:
1718:
1715:
1700:
1699:
1698:
1684:
1664:
1635:
1632:
1616:
1592:
1584:
1580:
1576:
1573:
1570:
1567:
1563:
1554:
1553:
1552:
1551:
1547:
1531:
1528:
1525:
1522:
1517:
1513:
1504:
1496:
1495:
1494:
1491:
1472:
1468:
1464:
1459:
1455:
1451:
1446:
1442:
1434:
1418:
1414:
1410:
1405:
1401:
1397:
1392:
1388:
1380:
1364:
1360:
1356:
1351:
1347:
1343:
1338:
1334:
1326:
1325:
1324:
1307:
1304:
1299:
1295:
1287:
1273:
1270:
1265:
1261:
1253:
1252:
1251:
1250:
1229:
1226:
1223:
1219:
1215:
1210:
1207:
1204:
1200:
1196:
1191:
1187:
1179:
1178:
1177:
1175:
1171:
1161:
1145:
1141:
1120:
1117:
1097:
1094:
1071:
1063:
1059:
1055:
1052:
1044:
1040:
1036:
1033:
1028:
1025:
1022:
1018:
1010:
1009:
1008:
1006:
996:
979:
972:
971:
970:
964:
945:
942:
939:
936:
929:
928:
927:
910:
907:
904:
901:
891:
885:
882:
879:
873:
870:
867:
864:
861:
854:
853:
852:
850:
835:
813:
805:
801:
797:
789:
786:
763:
760:
757:
754:
739:
736:
733:
729:
725:
722:
719:
714:
711:
708:
704:
700:
695:
692:
689:
685:
681:
678:
672:
669:
664:
660:
652:
651:
650:
645:
641:
636:
633:
631:
630:initial value
613:
609:
588:
585:
580:
576:
548:
542:
539:
531:
528:
521:
520:
519:
502:
499:
496:
493:
478:
475:
472:
468:
464:
461:
455:
452:
447:
443:
435:
434:
433:
431:
427:
417:
415:
411:
407:
403:
398:
384:
376:
371:
369:
365:
364:Taylor series
361:
345:
338:depending on
337:
321:
313:
297:
294:
286:
270:
248:
245:
242:
238:
234:
229:
226:
223:
219:
215:
210:
206:
197:
181:
173:
165:
160:
146:
138:
122:
102:
82:
62:
54:
51:th term of a
38:
30:
26:
22:
8598:
8576:
8531:
8527:
8496:. Retrieved
8489:the original
8484:
8451:
8432:. Springer.
8428:
8422:Cull, Paul;
8415:the original
8409:
8389:
8371:
8347:
8329:
8320:
8311:
8304:Bibliography
8280:
8265:
8245:
8227:
8219:
8214:
8206:
8201:
8176:. Retrieved
8156:
8137:
8128:
8022:
8004:
7632:
7522:
7507:
7447:
7349:
7326:
7303:
7288:
7269:
7192:
7015:logistic map
7012:
7001:
6993:Applications
6984:
6858:
6673:
6549:
6532:logistic map
6505:
6475:
6369:
6255:
6250:
6248:
6072:
6008:
5889:
5796:
5770:
5641:
5492:
5314:
5276:
5216:
5093:
5036:is given by
5035:
4937:
4806:
4185:
4093:
3656:
3558:
3532:
3490:
3471:
3467:
3455:
3452:
3369:
3282:
3259:
3249:relates the
3226:
3224:
3074:
2942:and one has
2884:
2879:
2701:
2699:
2661:
2599:
2441:
2364:
2357:
2229:
2227:
2099:
1989:
1818:
1612:
1500:
1492:
1489:
1322:
1246:
1167:
1086:
1005:logistic map
1002:
999:Logistic map
994:
962:
960:
925:
846:
778:
643:
639:
637:
634:
629:
563:
517:
425:
423:
399:
372:
163:
161:
136:
24:
18:
7525:comb filter
7026:for a host-
7020:populations
6555:numerically
5777:eigenvalues
5301:solutions.
5287:P-recursive
5285:are called
4946:, and many
3255:derivatives
1007:defined by
640:first order
404:, that is,
334:. As well,
21:mathematics
8625:Categories
8528:Anal. Appl
8498:2014-08-07
8444:chapter 7.
8178:2010-10-19
8115:References
8013:See also:
7279:univoltine
6390:appearing
5291:polynomial
2240:that maps
1899:factorials
420:Definition
8600:MathWorld
8583:EMS Press
8541:1101.4371
8120:Footnotes
8058:Recursion
8009:Economics
7982:−
7965:α
7953:−
7933:α
7929:−
7926:α
7904:α
7901:−
7850:−
7839:α
7828:−
7814:α
7811:−
7799:α
7780:α
7777:−
7735:α
7610:−
7599:α
7580:α
7577:−
7527:of delay
7476:
7295:algorithm
7157:−
7149:−
7078:−
7060:λ
6989:article.
6844:…
6489:∗
6442:∗
6343:∘
6340:⋯
6337:∘
6331:∘
6218:∗
6168:∗
6121:∗
6094:∗
6079:converges
6047:−
5963:∗
5870:∗
5862:−
5854:−
5829:∗
5821:−
5744:λ
5730:−
5727:⋯
5724:−
5716:−
5709:λ
5695:−
5687:−
5680:λ
5666:−
5657:λ
5619:−
5595:⋯
5584:−
5555:−
5484:Stability
5177:−
5152:−
5132:−
5115:−
5016:−
5005:−
4911:→
4757:∏
4735:−
4717:∑
4675:−
4657:∏
4587:∏
4565:−
4547:∑
4509:−
4491:∏
4426:∏
4404:−
4386:∑
4369:−
4340:−
4313:−
4295:∑
4243:∏
4213:−
4150:−
4132:∏
4049:∏
4011:−
3993:∏
3976:−
3943:∏
3870:∏
3820:∏
3791:−
3758:∏
3684:−
3636:≠
3422:−
3323:Δ
3298:Δ
3245:of order
3233:of order
3185:Δ
3156:⋯
3140:Δ
3052:−
3000:−
2977:∑
2954:Δ
2922:−
2915:Δ
2911:∘
2908:Δ
2896:Δ
2830:−
2792:Δ
2759:Δ
2753:Δ
2741:Δ
2738:∘
2735:Δ
2717:Δ
2680:Δ
2638:∈
2556:Δ
2530:Δ
2496:Δ
2473:Δ
2450:Δ
2414:−
2379:Δ
2328:−
2289:Δ
2256:Δ
2246:functions
2242:sequences
2193:−
2062:−
2041:−
1960:−
1786:−
1756:−
1745:−
1577:−
1571:−
1227:−
1208:−
1056:−
883:−
874:⋅
849:factorial
843:Factorial
811:→
798:×
787:φ
758:≥
737:−
723:…
712:−
693:−
673:φ
586:∈
546:→
540:×
529:φ
476:−
456:φ
246:−
227:−
8558:28828175
8330:SIAM Rev
8277:(2004).
8243:(1989).
8187:cite web
8169:Archived
8036:See also
7293:. If an
7028:parasite
6575:′
6540:tent map
6430:′
6206:′
5642:has the
5295:rational
5098:, while
2602:sequence
2548:and not
2238:operator
1160:varies.
838:Examples
430:sequence
53:sequence
29:equation
8631:Algebra
8585:, 2001
8338:2027658
8028:, real
7275:ecology
6508:chaotic
3539:Solving
3278:inverse
518:where
360:special
174:of the
8556:
8464:
8460:–568.
8436:
8396:
8380:
8366:, and
8336:
8289:
8253:
8148:
7727:, and
7633:where
7516:(IIR)
6841:
6794:
6791:
6747:
6744:
6624:
6621:
6538:, and
6476:where
5773:stable
4186:Then
3253:first
2600:Given
2275:, as
2236:is an
1544:; the
779:where
410:tuples
166:, the
27:is an
8554:S2CID
8536:arXiv
8492:(PDF)
8481:(PDF)
8334:JSTOR
8172:(PDF)
8165:(PDF)
8005:etc.
7193:with
6674:with
6506:In a
6370:with
6184:unity
5781:unity
4094:Let
1490:etc.
1247:with
644:order
137:order
8615:OEIS
8462:ISBN
8434:ISBN
8394:ISBN
8378:ISBN
8287:ISBN
8251:ISBN
8193:link
8146:ISBN
8017:and
7547:is:
7448:the
7013:The
6455:<
6287:>
6276:for
6231:<
5277:the
5094:the
3461:and
2708:is
2700:The
2660:the
2465:and
2228:The
905:>
847:The
497:>
23:, a
8546:doi
8458:551
8030:GDP
7508:In
7467:log
5297:or
2672:is
2668:of
897:for
750:for
489:for
412:of
370:).
314:of
162:In
19:In
8627::
8597:.
8581:,
8575:,
8552:.
8544:.
8532:10
8530:.
8483:.
8370:.
8362:,
8358:,
8273:;
8239:;
8235:;
8189:}}
8185:{{
8167:.
7680:,
7520:.
7500:.
7347:.
7267:.
6542:.
6534:,
6325::=
6234:1.
5930:,
5756:0.
5513:,
5480:.
5293:,
3535:.
3225:A
2362:.
1308:1.
969:)
946:1.
632:.
424:A
416:.
397:.
198:,
8612:.
8603:.
8560:.
8548::
8538::
8520:.
8510:.
8501:.
8470:.
8442:.
8402:.
8340:.
8295:.
8259:.
8195:)
8181:.
7988:T
7985:2
7979:t
7975:y
7969:2
7961:+
7956:T
7950:t
7946:x
7942:)
7937:2
7923:(
7920:+
7915:t
7911:x
7907:)
7898:1
7895:(
7892:=
7887:t
7883:y
7861:)
7856:T
7853:2
7847:t
7843:y
7836:+
7831:T
7825:t
7821:x
7817:)
7808:1
7805:(
7802:(
7796:+
7791:t
7787:x
7783:)
7774:1
7771:(
7768:=
7763:t
7759:y
7715:t
7693:t
7689:y
7668:t
7646:t
7642:x
7618:,
7613:T
7607:t
7603:y
7596:+
7591:t
7587:x
7583:)
7574:1
7571:(
7568:=
7563:t
7559:y
7535:T
7488:)
7485:)
7482:n
7479:(
7471:2
7463:(
7460:O
7431:2
7427:/
7423:n
7419:c
7415:+
7412:1
7409:=
7404:n
7400:c
7378:1
7375:=
7370:1
7366:c
7335:n
7312:n
7255:t
7233:t
7229:P
7206:t
7202:N
7178:,
7175:)
7168:t
7164:P
7160:a
7153:e
7146:1
7143:(
7138:t
7134:N
7130:=
7125:1
7122:+
7119:t
7115:P
7089:t
7085:P
7081:a
7074:e
7068:t
7064:N
7057:=
7052:1
7049:+
7046:t
7042:N
6970:h
6967:n
6964:+
6959:0
6955:t
6951:=
6946:n
6942:t
6938:,
6935:)
6930:n
6926:y
6922:,
6917:n
6913:t
6909:(
6906:f
6903:h
6900:+
6895:n
6891:y
6887:=
6882:1
6879:+
6876:n
6872:y
6838:,
6835:)
6832:h
6829:2
6826:+
6821:0
6817:t
6813:(
6810:y
6807:=
6802:2
6798:y
6788:,
6785:)
6782:h
6779:+
6774:0
6770:t
6766:(
6763:y
6760:=
6755:1
6751:y
6741:,
6738:)
6733:0
6729:t
6725:(
6722:y
6719:=
6714:0
6710:y
6686:h
6659:,
6654:0
6650:y
6646:=
6643:)
6638:0
6634:t
6630:(
6627:y
6618:,
6615:)
6612:)
6609:t
6606:(
6603:y
6600:,
6597:t
6594:(
6591:f
6588:=
6585:)
6582:t
6579:(
6572:y
6518:x
6485:x
6461:,
6458:1
6451:|
6447:)
6438:x
6434:(
6427:g
6422:|
6398:k
6378:f
6355:)
6352:x
6349:(
6346:f
6334:f
6328:f
6322:)
6319:x
6316:(
6313:g
6290:1
6284:k
6264:k
6251:f
6227:|
6223:)
6214:x
6210:(
6203:f
6198:|
6164:x
6143:f
6117:x
6090:x
6058:.
6055:)
6050:1
6044:n
6040:x
6036:(
6033:f
6030:=
6025:n
6021:x
5985:A
5959:x
5938:x
5918:A
5898:x
5875:]
5866:x
5857:1
5851:t
5847:x
5843:[
5840:A
5837:=
5834:]
5825:x
5816:t
5812:x
5808:[
5753:=
5748:0
5738:d
5734:c
5719:2
5713:d
5703:2
5699:c
5690:1
5684:d
5674:1
5670:c
5661:d
5627:,
5622:d
5616:n
5612:a
5606:d
5602:c
5598:+
5592:+
5587:2
5581:n
5577:a
5571:2
5567:c
5563:+
5558:1
5552:n
5548:a
5542:1
5538:c
5534:=
5529:n
5525:a
5501:d
5466:t
5462:x
5439:t
5435:x
5412:t
5408:w
5383:d
5380:+
5375:t
5371:w
5367:c
5362:b
5359:+
5354:t
5350:w
5346:a
5339:=
5334:1
5331:+
5328:t
5324:w
5262:)
5259:z
5256:;
5253:b
5250:,
5247:n
5244:(
5241:M
5238:=
5233:n
5229:M
5202:0
5199:=
5194:1
5191:+
5188:n
5184:M
5180:n
5172:n
5168:M
5164:)
5161:z
5158:+
5155:b
5149:n
5146:2
5143:(
5140:+
5135:1
5129:n
5125:M
5121:)
5118:n
5112:b
5109:(
5079:,
5076:)
5073:z
5070:(
5065:n
5061:J
5057:=
5052:n
5048:J
5019:1
5013:n
5009:J
5000:n
4996:J
4990:z
4986:n
4983:2
4977:=
4972:1
4969:+
4966:n
4962:J
4914:0
4908:h
4886:h
4883:n
4879:g
4875:h
4872:+
4867:n
4863:a
4859:)
4854:h
4851:n
4847:f
4843:h
4840:+
4837:1
4834:(
4831:=
4826:1
4823:+
4820:n
4816:a
4791:)
4782:k
4778:f
4772:m
4767:0
4764:=
4761:k
4750:m
4746:g
4738:1
4732:n
4727:0
4724:=
4721:m
4713:+
4708:0
4704:A
4699:(
4694:)
4688:k
4684:f
4678:1
4672:n
4667:0
4664:=
4661:k
4652:(
4648:=
4643:n
4639:a
4612:k
4608:f
4602:m
4597:0
4594:=
4591:k
4580:m
4576:g
4568:1
4562:n
4557:0
4554:=
4551:m
4543:+
4538:0
4534:A
4530:=
4522:k
4518:f
4512:1
4506:n
4501:0
4498:=
4495:k
4484:n
4480:a
4451:k
4447:f
4441:m
4436:0
4433:=
4430:k
4419:m
4415:g
4407:1
4401:n
4396:0
4393:=
4390:m
4382:=
4377:0
4373:A
4364:n
4360:A
4356:=
4353:)
4348:m
4344:A
4335:1
4332:+
4329:m
4325:A
4321:(
4316:1
4310:n
4305:0
4302:=
4299:m
4268:k
4264:f
4258:n
4253:0
4250:=
4247:k
4236:n
4232:g
4226:=
4221:n
4217:A
4208:1
4205:+
4202:n
4198:A
4171:,
4163:k
4159:f
4153:1
4147:n
4142:0
4139:=
4136:k
4125:n
4121:a
4115:=
4110:n
4106:A
4074:k
4070:f
4064:n
4059:0
4056:=
4053:k
4042:n
4038:g
4032:=
4024:k
4020:f
4014:1
4008:n
4003:0
4000:=
3997:k
3986:n
3982:a
3968:k
3964:f
3958:n
3953:0
3950:=
3947:k
3936:1
3933:+
3930:n
3926:a
3895:k
3891:f
3885:n
3880:0
3877:=
3874:k
3863:n
3859:g
3853:=
3845:k
3841:f
3835:n
3830:0
3827:=
3824:k
3812:n
3808:a
3802:n
3798:f
3783:k
3779:f
3773:n
3768:0
3765:=
3762:k
3751:1
3748:+
3745:n
3741:a
3715:n
3711:g
3707:=
3702:n
3698:a
3692:n
3688:f
3679:1
3676:+
3673:n
3669:a
3642:,
3639:0
3631:n
3627:f
3622:,
3617:n
3613:g
3609:+
3604:n
3600:a
3594:n
3590:f
3586:=
3581:1
3578:+
3575:n
3571:a
3519:n
3499:n
3438:,
3433:n
3429:a
3425:8
3417:1
3414:+
3411:n
3407:a
3403:4
3400:=
3395:2
3392:+
3389:n
3385:a
3381:3
3355:0
3352:=
3347:n
3343:a
3339:7
3336:+
3331:n
3327:a
3320:2
3317:+
3312:n
3308:a
3302:2
3294:3
3274:k
3270:k
3266:k
3262:k
3251:k
3247:k
3239:k
3235:k
3210:.
3207:)
3202:n
3198:a
3194:(
3189:k
3178:)
3173:k
3170:k
3165:(
3159:+
3153:+
3148:n
3144:a
3134:)
3129:1
3126:k
3121:(
3115:+
3110:n
3106:a
3102:=
3097:k
3094:+
3091:n
3087:a
3060:.
3055:t
3049:k
3046:+
3043:n
3039:a
3032:)
3027:t
3024:k
3019:(
3011:t
3007:)
3003:1
2997:(
2992:k
2987:0
2984:=
2981:t
2973:=
2968:n
2964:a
2958:k
2930:,
2925:1
2919:k
2905:=
2900:k
2882:k
2865:.
2860:n
2856:a
2852:+
2847:1
2844:+
2841:n
2837:a
2833:2
2825:2
2822:+
2819:n
2815:a
2811:=
2806:n
2802:a
2796:2
2768:.
2765:)
2762:a
2756:(
2750:=
2747:a
2744:)
2732:(
2729:=
2726:a
2721:2
2686:.
2683:a
2670:a
2648:,
2642:N
2635:n
2631:)
2625:n
2621:a
2617:(
2614:=
2611:a
2586:.
2581:n
2577:a
2536:,
2533:a
2520:n
2504:n
2500:a
2476:a
2453:f
2427:.
2422:n
2418:a
2409:1
2406:+
2403:n
2399:a
2395:=
2390:n
2386:)
2382:a
2376:(
2343:.
2340:)
2337:x
2334:(
2331:f
2325:)
2322:1
2319:+
2316:x
2313:(
2310:f
2307:=
2304:)
2301:x
2298:(
2295:)
2292:f
2286:(
2259:,
2208:,
2202:)
2196:k
2190:n
2186:n
2181:(
2175:=
2169:)
2164:k
2161:n
2156:(
2133:1
2130:=
2124:)
2119:0
2116:n
2111:(
2085:,
2082:k
2078:/
2074:)
2071:1
2068:+
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2056:(
2050:)
2044:1
2038:k
2034:n
2029:(
2023:=
2017:)
2012:k
2009:n
2004:(
1975:.
1969:!
1966:)
1963:k
1957:n
1954:(
1951:!
1948:k
1943:!
1940:n
1934:=
1928:)
1923:k
1920:n
1915:(
1881:1
1878:=
1871:)
1866:n
1863:n
1858:(
1851:=
1844:)
1839:0
1836:n
1831:(
1804:,
1798:)
1793:k
1789:1
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1777:(
1771:+
1765:)
1759:1
1753:k
1748:1
1742:n
1736:(
1730:=
1724:)
1719:k
1716:n
1711:(
1685:n
1665:k
1641:)
1636:k
1633:n
1628:(
1593:.
1585:2
1581:t
1574:t
1568:1
1564:t
1532:1
1529:+
1526:t
1523:=
1518:2
1514:t
1473:2
1469:F
1465:+
1460:3
1456:F
1452:=
1447:4
1443:F
1419:1
1415:F
1411:+
1406:2
1402:F
1398:=
1393:3
1389:F
1365:0
1361:F
1357:+
1352:1
1348:F
1344:=
1339:2
1335:F
1305:=
1300:1
1296:F
1274:0
1271:=
1266:0
1262:F
1230:2
1224:n
1220:F
1216:+
1211:1
1205:n
1201:F
1197:=
1192:n
1188:F
1146:0
1142:x
1121:,
1118:r
1098:.
1095:r
1072:,
1069:)
1064:n
1060:x
1053:1
1050:(
1045:n
1041:x
1037:r
1034:=
1029:1
1026:+
1023:n
1019:x
980:n
967:n
943:=
940:!
937:0
911:,
908:0
902:n
892:!
889:)
886:1
880:n
877:(
871:n
868:=
865:!
862:n
832:k
828:k
814:X
806:k
802:X
794:N
790::
764:,
761:k
755:n
745:)
740:k
734:n
730:u
726:,
720:,
715:2
709:n
705:u
701:,
696:1
690:n
686:u
682:,
679:n
676:(
670:=
665:n
661:u
647:k
614:0
610:u
589:X
581:0
577:u
566:X
549:X
543:X
536:N
532::
503:,
500:0
494:n
484:)
479:1
473:n
469:u
465:,
462:n
459:(
453:=
448:n
444:u
385:n
346:n
322:n
298:.
295:n
271:k
249:2
243:n
239:F
235:+
230:1
224:n
220:F
216:=
211:n
207:F
182:k
168:n
147:k
123:k
103:n
83:k
63:k
39:n
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