2156:(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
7365:. 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
8034:
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
3921:
4100:
4813:
4638:
4477:
5777:
785:
3231:
8011:
7882:
7028:
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
6865:
3081:
3479:
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.
3467:
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
524:
2789:
1825:
5648:
4294:
3744:
4192:
3929:
932:
6991:
3663:
4644:
2106:
6680:
3376:
2886:
2951:
1996:
7639:
835:
6376:
5408:
5223:
5042:
1902:
570:
7114:
4909:
3502:
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
2229:
2669:
7199:
4483:
2448:
4300:
2364:
3459:
3738:
1093:
1614:
6482:
6255:
1253:
272:
6079:
1496:
1442:
1388:
7338:
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
1666:
7509:
7454:
5100:
5283:
2154:
2527:
610:
2707:
2557:
1553:
5662:
2497:
2474:
2280:
1329:
666:
7756:
7399:
3092:
2577:
1295:
6311:
4935:
2607:
7716:
7669:
7256:
7229:
6512:
6191:
6144:
6117:
5986:
5489:
5462:
5435:
1169:
967:
637:
5896:
1142:
1119:
319:
7736:
7689:
7556:
7356:
7333:
7276:
6707:
6539:
6419:
6399:
6285:
6164:
6006:
5959:
5939:
5919:
5522:
5301:
3540:
3520:
1706:
1686:
1001:
406:
367:
343:
292:
203:
168:
144:
124:
104:
84:
60:
443:
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
7888:
3560:
295:
8438:
6260:
A nonlinear recurrence could have multiple fixed points, in which case some fixed points may be locally stable and others locally unstable; for continuous
6541:
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
8203:
7764:
6715:
8628:
index to a few thousand examples of linear recurrences, sorted by order (number of terms) and signature (vector of values of the constant coefficients)
2959:
6565:
8488:
8083:
8419:
449:
8160:
2722:
1714:
5530:
3916:{\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}}}}
4203:
4095:{\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}}}}
3291:
of the other, and the sequences that are solution of the difference equation are exactly those that satisfies the recurrence relation.
4111:
868:
8476:
8448:
8408:
8392:
8301:
8265:
8113:
6876:
4950:
4808:{\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)}
7523:, recurrence relations can model feedback in a system, where outputs at one time become inputs for future time. They thus arise in
3576:
2007:
6313:. Such a cycle is stable, meaning that it attracts a set of initial conditions of positive measure, if the composite function
5289:
3300:
2797:
6562:
3253:
2901:
1918:
7564:
6996:
Systems of linear first order differential equations can be discretized exactly analytically using the methods shown in the
793:
6319:
5329:
5115:
212:
8646:
8593:
4967:
4941:
with variable coefficients; the sum becomes an integral, and the product becomes the exponential function of an integral.
8179:
6577:
1833:
170:
numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation.
8088:
8073:
8029:
7309:
5321:
4938:
3469:
535:
8588:
8035:
some agents' actions depend on lagged variables. The model would then be solved for current values of key variables (
7047:
4821:
4633:{\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}}}}
7034:
5309:
4472:{\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}}}}
2617:
2159:
8383:
8078:
7281:
7120:
5803:
3473:
2382:
8108:
7524:
7520:
5654:
2292:
8103:
3387:
3674:
1024:
5464:
which itself evolves linearly. Then standard methods can be used to solve the linear difference equation in
3570:
Moreover, for the general first-order non-homogeneous linear recurrence relation with variable coefficients:
1568:
8537:
Wang, Xiang-Sheng; Wong, Roderick (2012). "Asymptotics of orthogonal polynomials via recurrence relations".
7362:
1904:. Using this formula to compute the values of all binomial coefficients generates an infinite array called
1193:
8651:
8499:
8118:
8063:
7301:
5305:
4954:
2256:
2248:
1908:. The same values can also be computed directly by a different formula that is not a recurrence, but uses
412:
322:
8583:
1187:
relation with constant coefficients (see below). The
Fibonacci sequence is defined using the recurrence
6569:
6546:
6427:
6203:
6026:
1448:
1394:
1340:
646:
It is easy to modify the definition for getting sequences starting from the term of index 1 or higher.
1630:
8370:
8025:
7466:
7405:
5293:
5053:
1905:
1625:
385:
346:
8425:
5234:
8098:
7014:
6194:
6089:
5791:
3491:
2283:
2114:
1556:
2502:
8564:
8546:
8344:
8251:
8093:
8053:
5772:{\displaystyle \lambda ^{d}-c_{1}\lambda ^{d-1}-c_{2}\lambda ^{d-2}-\cdots -c_{d}\lambda ^{0}=0.}
5297:
3483:
780:{\displaystyle u_{n}=\varphi (n,u_{n-1},u_{n-2},\ldots ,u_{n-k})\quad {\text{for}}\quad n\geq k,}
378:
7758:
controls how much of the delayed signal is fed back into the output. From this we can see that
3226:{\displaystyle a_{n+k}=a_{n}+{k \choose 1}\Delta a_{n}+\cdots +{k \choose k}\Delta ^{k}(a_{n}).}
1513:
582:
298:, because the coefficients of the linear function (1 and 1) are constants that do not depend on
8468:
2686:
2536:
1519:
8602:
8472:
8444:
8404:
8388:
8297:
8285:
8261:
8247:
8197:
8156:
8058:
6686:
3487:
2479:
2456:
2370:
2262:
1560:
1301:
1259:
1184:
370:
7741:
7371:
7315:
A simple example is the time an algorithm takes to find an element in an ordered vector with
5790:(i.e., the roots of the characteristic equation), whether real or complex, are all less than
3494:
for a unification of the theory of difference equations with that of differential equations.
2562:
1267:
8641:
8556:
8460:
8374:
8366:
7018:
6290:
6085:
5783:
4958:
4914:
3288:
2582:
1180:
416:
206:
7694:
7647:
7234:
7207:
6490:
6169:
6122:
6095:
5964:
5467:
5440:
5413:
1147:
943:
615:
8281:
7460:
5106:
182:
5786:, meaning that the iterates converge asymptotically to a fixed value, if and only if the
8605:
5814:
1124:
1101:
301:
8378:
8290:
8006:{\displaystyle y_{t}=(1-\alpha )x_{t}+(\alpha -\alpha ^{2})x_{t-T}+\alpha ^{2}y_{t-2T}}
7721:
7674:
7541:
7528:
7341:
7318:
7261:
7013:
Some of the best-known difference equations have their origins in the attempt to model
6997:
6692:
6524:
6404:
6384:
6270:
6149:
6009:
5991:
5944:
5924:
5904:
5507:
3525:
3505:
1691:
1671:
986:
424:
391:
352:
328:
277:
188:
153:
129:
109:
89:
69:
45:
8635:
8461:
8036:
374:
294:
is two and the linear function merely adds the two previous terms. This example is a
8568:
17:
8528:
8434:
8243:
7025:
6542:
6518:
3566:
Solving first-order non-homogeneous recurrence relations with variable coefficients
1015:
7877:{\displaystyle y_{t}=(1-\alpha )x_{t}+\alpha ((1-\alpha )x_{t-T}+\alpha y_{t-2T})}
6860:{\displaystyle y_{0}=y(t_{0}),\ \ y_{1}=y(t_{0}+h),\ \ y_{2}=y(t_{0}+2h),\ \dots }
2001:
The binomial coefficients can also be computed with a uni-dimensional recurrence:
8255:
8152:
3076:{\displaystyle \Delta ^{k}a_{n}=\sum _{t=0}^{k}(-1)^{t}{\binom {k}{t}}a_{n+k-t}.}
8172:
7535:
6568:, one typically encounters a recurrence relation. For example, when solving the
5787:
31:
8518:
8489:"Using generating functions to solve linear inhomogeneous recurrence equations"
8560:
7030:
3476:
for example of uses of "difference equation" instead of "recurrence relation"
3265:
8620:
321:
For these recurrences, one can express the general term of the sequence as a
8610:
8068:
7305:
7289:
1909:
859:
7288:. These and other difference equations are particularly suited to modeling
4949:
Many homogeneous linear recurrence relations may be solved by means of the
1624:
A simple example of a multidimensional recurrence relation is given by the
66:
of numbers is equal to some combination of the previous terms. Often, only
5300:. For these specific recurrence equations algorithms are known which find
3271:
The two above relations allow transforming a recurrence relation of order
1516:, which involves powers of the two roots of the characteristic polynomial
7038:
6550:
3464:
in the sense that the two equations are satisfied by the same sequences.
2612:
2252:
440:
63:
39:
8339:
Fillmore, Jay P.; Marx, Morris L. (1968). "Linear recursive sequences".
519:{\displaystyle u_{n}=\varphi (n,u_{n-1})\quad {\text{for}}\quad n>0,}
8348:
7285:
2784:{\displaystyle \Delta ^{2}a=(\Delta \circ \Delta )a=\Delta (\Delta a).}
86:
previous terms of the sequence appear in the equation, for a parameter
7308:
is designed so that it will break a problem into smaller subproblems (
1820:{\displaystyle {\binom {n}{k}}={\binom {n-1}{k-1}}+{\binom {n-1}{k}},}
5643:{\displaystyle a_{n}=c_{1}a_{n-1}+c_{2}a_{n-2}+\cdots +c_{d}a_{n-d},}
8357:
4289:{\displaystyle A_{n+1}-A_{n}={\frac {g_{n}}{\prod _{k=0}^{n}f_{k}}}}
3252:
first differences of a sequence or a function, in the same way as a
2376:
When using the index notation for sequences, the definition becomes
8551:
6267:
A nonlinear recurrence relation could also have a cycle of period
579:
is a set to which the elements of a sequence must belong. For any
420:
7021:
were once used as a model for the growth of a rabbit population.
1512:
The recurrence can be solved by methods described below yielding
8625:
8043:, etc.) in terms of past and current values of other variables.
8496:
Proc. Int. Conf. Simulation, Modelling and
Optimization, SMO'06
4187:{\displaystyle A_{n}={\frac {a_{n}}{\prod _{k=0}^{n-1}f_{k}}},}
927:{\displaystyle n!=n\cdot (n-1)!\quad {\text{for}}\quad n>0,}
8040:
6986:{\displaystyle \,y_{n+1}=y_{n}+hf(t_{n},y_{n}),t_{n}=t_{0}+nh}
4953:. Special cases of these lead to recurrence relations for the
3555:
Solving linear recurrence relations with constant coefficients
8399:
Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1994).
3658:{\displaystyle a_{n+1}=f_{n}a_{n}+g_{n},\qquad f_{n}\neq 0,}
7312:), its running time is described by a recurrence relation.
7300:
Recurrence relations are also of fundamental importance in
2101:{\displaystyle {\binom {n}{k}}={\binom {n}{k-1}}(n-k+1)/k,}
377:
whose coefficients satisfy such a recurrence relation (see
205:
previous terms. A famous example is the recurrence for the
8143:
Jacobson, Nathan, Basic
Algebra 2 (2nd ed.), § 0.4. pg 16.
2231:
all involved integers are smaller than the final result).
1708:
elements. They can be computed by the recurrence relation
1504:
We obtain the sequence of
Fibonacci numbers, which begins
6421:
times is locally stable according to the same criterion:
6264:
two adjacent fixed points cannot both be locally stable.
5988:
if and only if all eigenvalues of the transition matrix
5326:
A first order rational difference equation has the form
5294:
linear difference equations with polynomial coefficients
3371:{\displaystyle 3\Delta ^{2}a_{n}+2\Delta a_{n}+7a_{n}=0}
2881:{\displaystyle \Delta ^{2}a_{n}=a_{n+2}-2a_{n+1}+a_{n}.}
411:
The concept of a recurrence relation can be extended to
8401:
Concrete
Mathematics: A Foundation for Computer Science
7284:
are a form of recurrence relation important to spatial
4945:
Solving general homogeneous linear recurrence relations
2946:{\displaystyle \Delta ^{k}=\Delta \circ \Delta ^{k-1},}
1991:{\displaystyle {\binom {n}{k}}={\frac {n!}{k!(n-k)!}}.}
369:
are also important, because many common elementary and
8519:"Difference and Functional Equations: Exact Solutions"
7634:{\displaystyle y_{t}=(1-\alpha )x_{t}+\alpha y_{t-T},}
5353:
2162:
2117:
1865:
1838:
1635:
830:{\displaystyle \varphi :\mathbb {N} \times X^{k}\to X}
7891:
7767:
7744:
7724:
7697:
7677:
7650:
7567:
7544:
7469:
7408:
7374:
7344:
7321:
7264:
7237:
7210:
7123:
7050:
6879:
6718:
6695:
6580:
6527:
6493:
6430:
6407:
6387:
6371:{\displaystyle g(x):=f\circ f\circ \cdots \circ f(x)}
6322:
6293:
6273:
6206:
6172:
6152:
6125:
6098:
6029:
5994:
5967:
5947:
5927:
5907:
5817:
5665:
5533:
5510:
5470:
5443:
5416:
5403:{\displaystyle w_{t+1}={\tfrac {aw_{t}+b}{cw_{t}+d}}}
5332:
5237:
5218:{\displaystyle (b-n)M_{n-1}+(2n-b+z)M_{n}-nM_{n+1}=0}
5118:
5056:
4970:
4917:
4824:
4647:
4486:
4303:
4206:
4114:
3932:
3747:
3677:
3579:
3528:
3508:
3390:
3303:
3095:
2962:
2904:
2800:
2725:
2689:
2620:
2585:
2565:
2539:
2505:
2482:
2459:
2385:
2295:
2265:
2010:
1921:
1836:
1717:
1694:
1674:
1633:
1571:
1522:
1451:
1397:
1343:
1304:
1270:
1196:
1150:
1127:
1121:
The behavior of the sequence depends dramatically on
1104:
1027:
989:
946:
871:
796:
669:
618:
585:
538:
452:
394:
355:
331:
304:
280:
215:
191:
156:
132:
112:
92:
72:
48:
5961:
converges asymptotically to the steady state vector
5037:{\displaystyle J_{n+1}={\frac {2n}{z}}J_{n}-J_{n-1}}
841:
consecutive elements of the sequence. In this case,
8387:, Second Edition. MIT Press and McGraw-Hill, 1990.
6675:{\displaystyle y'(t)=f(t,y(t)),\ \ y(t_{0})=y_{0},}
1912:, multiplication and division, not just additions:
1897:{\displaystyle {\tbinom {n}{0}}={\tbinom {n}{n}}=1}
845:initial values are needed for defining a sequence.
8289:
8005:
7876:
7750:
7730:
7710:
7683:
7663:
7633:
7550:
7534:For example, the equation for a "feedforward" IIR
7503:
7448:
7393:
7350:
7327:
7270:
7250:
7223:
7193:
7108:
6985:
6859:
6701:
6674:
6533:
6506:
6476:
6413:
6393:
6370:
6305:
6279:
6249:
6185:
6158:
6138:
6111:
6073:
6000:
5980:
5953:
5933:
5913:
5890:
5798:Stability of linear first-order matrix recurrences
5771:
5642:
5516:
5483:
5456:
5437:as a nonlinear transformation of another variable
5429:
5402:
5277:
5217:
5094:
5036:
4929:
4903:
4807:
4632:
4471:
4288:
4186:
4094:
3915:
3732:
3657:
3534:
3514:
3453:
3370:
3279:, and, conversely, a difference equation of order
3225:
3075:
2945:
2880:
2783:
2701:
2663:
2601:
2571:
2551:
2521:
2491:
2468:
2442:
2358:
2274:
2223:
2148:
2100:
1990:
1896:
1819:
1700:
1680:
1660:
1608:
1547:
1490:
1436:
1382:
1323:
1289:
1247:
1163:
1136:
1113:
1087:
995:
961:
926:
829:
779:
631:
604:
565:{\displaystyle \varphi :\mathbb {N} \times X\to X}
564:
518:
400:
361:
337:
313:
286:
266:
197:
162:
138:
118:
98:
78:
54:
8533:at EqWorld - The World of Mathematical Equations.
8523:at EqWorld - The World of Mathematical Equations.
5316:Solving first-order rational difference equations
3188:
3175:
3144:
3131:
3042:
3029:
2060:
2039:
2027:
2014:
1938:
1925:
1808:
1787:
1775:
1746:
1734:
1721:
7109:{\displaystyle N_{t+1}=\lambda N_{t}e^{-aP_{t}}}
4904:{\displaystyle a_{n+1}=(1+hf_{nh})a_{n}+hg_{nh}}
1334:Explicitly, the recurrence yields the equations
439:is an equation that expresses each element of a
384:Solving a recurrence relation means obtaining a
27:Pattern defining an infinite sequence of numbers
2224:{\textstyle {\binom {n}{k}}={\binom {n}{n-k}},}
347:linear recurrences with polynomial coefficients
8529:"Difference and Functional Equations: Methods"
8343:. Vol. 10, no. 3. pp. 324–353.
8323:An introduction to linear difference equations
6020:Consider the nonlinear first-order recurrence
6016:Stability of nonlinear first-order recurrences
5808:In the first-order matrix difference equation
2664:{\displaystyle a=(a_{n})_{n\in \mathbb {N} },}
974:linear recurrence with polynomial coefficients
8231:An Introduction to the Analysis of Algorithms
7194:{\displaystyle P_{t+1}=N_{t}(1-e^{-aP_{t}}),}
2443:{\displaystyle (\Delta a)_{n}=a_{n+1}-a_{n}.}
2235:Difference operator and difference equations
2212:
2191:
2179:
2166:
2134:
2121:
1881:
1868:
1854:
1841:
1651:
1638:
1179:The recurrence of order two satisfied by the
8:
5500:Stability of linear higher-order recurrences
5410:. Such an equation can be solved by writing
3561:Linear recurrence with constant coefficients
1508:0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
296:linear recurrence with constant coefficients
150:of the relation. If the values of the first
8440:Difference Equations: From Rabbits to Chaos
1014:An example of a recurrence relation is the
976:of order 1, with the simple polynomial (in
8467:(Fifth ed.). Prentice Hall. pp.
2359:{\displaystyle (\Delta f)(x)=f(x+1)-f(x).}
1183:is the canonical example of a homogeneous
8550:
8395:. Chapter 4: Recurrences, pp. 62–90.
8296:(Second ed.). Cambridge: MIT Press.
7988:
7978:
7959:
7946:
7924:
7896:
7890:
7856:
7834:
7800:
7772:
7766:
7743:
7723:
7702:
7696:
7676:
7655:
7649:
7616:
7600:
7572:
7566:
7543:
7480:
7468:
7436:
7432:
7413:
7407:
7379:
7373:
7343:
7320:
7263:
7242:
7236:
7215:
7209:
7177:
7166:
7147:
7128:
7122:
7098:
7087:
7077:
7055:
7049:
6968:
6955:
6939:
6926:
6904:
6885:
6880:
6878:
6830:
6811:
6783:
6764:
6742:
6723:
6717:
6694:
6663:
6647:
6579:
6526:
6498:
6492:
6460:
6451:
6431:
6429:
6406:
6386:
6321:
6292:
6272:
6236:
6227:
6207:
6205:
6177:
6171:
6151:
6130:
6124:
6103:
6097:
6053:
6034:
6028:
5993:
5972:
5966:
5946:
5926:
5906:
5879:
5860:
5838:
5825:
5816:
5757:
5747:
5722:
5712:
5693:
5683:
5670:
5664:
5625:
5615:
5590:
5580:
5561:
5551:
5538:
5532:
5509:
5475:
5469:
5448:
5442:
5421:
5415:
5384:
5363:
5352:
5337:
5331:
5242:
5236:
5197:
5181:
5138:
5117:
5074:
5061:
5055:
5022:
5009:
4990:
4975:
4969:
4916:
4892:
4876:
4860:
4829:
4823:
4791:
4781:
4770:
4759:
4753:
4741:
4730:
4717:
4697:
4681:
4670:
4652:
4646:
4621:
4611:
4600:
4589:
4583:
4571:
4560:
4547:
4531:
4515:
4504:
4493:
4487:
4485:
4460:
4450:
4439:
4428:
4422:
4410:
4399:
4386:
4373:
4357:
4338:
4319:
4308:
4302:
4277:
4267:
4256:
4245:
4239:
4230:
4211:
4205:
4172:
4156:
4145:
4134:
4128:
4119:
4113:
4083:
4073:
4062:
4051:
4045:
4033:
4017:
4006:
3995:
3989:
3977:
3967:
3956:
3939:
3933:
3931:
3904:
3894:
3883:
3872:
3866:
3854:
3844:
3833:
3821:
3811:
3804:
3792:
3782:
3771:
3754:
3748:
3746:
3724:
3711:
3701:
3682:
3676:
3668:there is also a nice method to solve it:
3640:
3626:
3613:
3603:
3584:
3578:
3527:
3522:-dimensional grids. Functions defined on
3507:
3454:{\displaystyle 3a_{n+2}=4a_{n+1}-8a_{n},}
3442:
3420:
3398:
3389:
3381:is equivalent to the recurrence relation
3356:
3340:
3321:
3311:
3302:
3211:
3198:
3187:
3174:
3172:
3157:
3143:
3130:
3128:
3119:
3100:
3094:
3052:
3041:
3028:
3026:
3020:
3001:
2990:
2977:
2967:
2961:
2928:
2909:
2903:
2869:
2850:
2828:
2815:
2805:
2799:
2730:
2724:
2688:
2652:
2651:
2644:
2634:
2619:
2590:
2584:
2564:
2538:
2513:
2504:
2481:
2458:
2431:
2412:
2399:
2384:
2294:
2264:
2211:
2190:
2188:
2178:
2165:
2163:
2161:
2133:
2120:
2118:
2116:
2087:
2059:
2038:
2036:
2026:
2013:
2011:
2009:
1947:
1937:
1924:
1922:
1920:
1880:
1867:
1864:
1853:
1840:
1837:
1835:
1807:
1786:
1784:
1774:
1745:
1743:
1733:
1720:
1718:
1716:
1693:
1673:
1650:
1637:
1634:
1632:
1594:
1572:
1570:
1527:
1521:
1482:
1469:
1456:
1450:
1428:
1415:
1402:
1396:
1374:
1361:
1348:
1342:
1309:
1303:
1275:
1269:
1233:
1214:
1201:
1195:
1155:
1149:
1144:but is stable when the initial condition
1126:
1103:
1073:
1054:
1032:
1026:
988:
945:
906:
870:
815:
804:
803:
795:
759:
743:
718:
699:
674:
668:
623:
617:
590:
584:
546:
545:
537:
498:
482:
457:
451:
393:
354:
330:
303:
279:
252:
233:
220:
214:
190:
155:
131:
111:
91:
71:
47:
8359:Linear Recursion and Fibonacci Sequences
3733:{\displaystyle a_{n+1}-f_{n}a_{n}=g_{n}}
2529:must be understood as the term of index
1088:{\displaystyle x_{n+1}=rx_{n}(1-x_{n}),}
8260:. Cambridge: Harvard University Press.
8136:
8084:Master theorem (analysis of algorithms)
5292:. Sequences which are the solutions of
1609:{\displaystyle {\frac {t}{1-t-t^{2}}}.}
8463:Mathematics for Economics and Business
8257:Recursive Methods in Economic Dynamics
8202:: CS1 maint: archived copy as title (
8195:
6557:Relationship to differential equations
3490:relate to differential equations. See
3086:This relation can be inverted, giving
862:is defined by the recurrence relation
612:, this defines a unique sequence with
4937:, we get the formula for first order
3294:For example, the difference equation
2259:to functions. It is commonly denoted
1248:{\displaystyle F_{n}=F_{n-1}+F_{n-2}}
267:{\displaystyle F_{n}=F_{n-1}+F_{n-2}}
7:
3275:into a difference equation of order
1668:, which count the ways of selecting
649:This defines recurrence relation of
2255:to sequences, and, more generally,
8498:. pp. 399–404. Archived from
8483:Chapter 9.1: Difference Equations.
8155:, Sui Sun Cheng, CRC Press, 2003,
6521:recurrence relation, the variable
6119:from points sufficiently close to
6008:(whether real or complex) have an
3486:relate to difference equations as
3333:
3308:
3283:into recurrence relation of order
3195:
3179:
3150:
3135:
3033:
2964:
2925:
2918:
2906:
2802:
2769:
2763:
2751:
2745:
2727:
2690:
2566:
2540:
2506:
2483:
2460:
2389:
2299:
2266:
2195:
2170:
2125:
2043:
2018:
1929:
1872:
1845:
1791:
1750:
1725:
1642:
25:
8114:Integration by reduction formulae
6477:{\displaystyle |g'(x^{*})|<1,}
6250:{\displaystyle |f'(x^{*})|<1.}
6074:{\displaystyle x_{n}=f(x_{n-1}).}
4951:generalized hypergeometric series
3248:is an equation that involves the
1491:{\displaystyle F_{4}=F_{3}+F_{2}}
1437:{\displaystyle F_{3}=F_{2}+F_{1}}
1383:{\displaystyle F_{2}=F_{1}+F_{0}}
639:as its first element, called the
8487:Minh, Tang; Van To, Tan (2006).
8421:Applied Econometric Times Series
3542:-grids can also be studied with
2791:A simple computation shows that
1661:{\displaystyle {\tbinom {n}{k}}}
8185:from the original on 2010-07-05
7504:{\displaystyle O(\log _{2}(n))}
7449:{\displaystyle c_{n}=1+c_{n/2}}
5504:The linear recurrence of order
5290:confluent hypergeometric series
5095:{\displaystyle J_{n}=J_{n}(z),}
4961:. For example, the solution to
3635:
911:
905:
764:
758:
503:
497:
8403:(2 ed.). Addison-Wesley.
8292:Recursive Macroeconomic Theory
7952:
7933:
7917:
7905:
7871:
7827:
7815:
7812:
7793:
7781:
7593:
7581:
7498:
7495:
7489:
7473:
7335:elements, in the worst case.
7185:
7153:
6945:
6919:
6845:
6823:
6795:
6776:
6748:
6735:
6653:
6640:
6625:
6622:
6616:
6604:
6595:
6589:
6563:ordinary differential equation
6461:
6457:
6444:
6432:
6365:
6359:
6332:
6326:
6237:
6233:
6220:
6208:
6065:
6046:
5885:
5853:
5844:
5818:
5278:{\displaystyle M_{n}=M(n,b;z)}
5272:
5254:
5174:
5153:
5131:
5119:
5086:
5080:
4921:
4869:
4844:
4363:
4331:
3217:
3204:
3017:
3007:
2775:
2766:
2754:
2742:
2641:
2627:
2396:
2386:
2350:
2344:
2335:
2323:
2314:
2308:
2305:
2296:
2149:{\textstyle {\binom {n}{0}}=1}
2084:
2066:
1976:
1964:
1079:
1060:
899:
887:
821:
755:
686:
556:
494:
469:
388:: a non-recursive function of
1:
7361:A better algorithm is called
6197:in absolute value: that is,
4939:linear differential equations
3287:. Each transformation is the
2369:It is thus a special case of
8424:(3 ed.). Archived from
8321:Batchelder, Paul M. (1967).
8153:Partial difference equations
8089:Circle points segments proof
8074:Recursion (computer science)
8030:simultaneous equations model
7231:representing the hosts, and
6709:, one calculates the values
5322:Rational difference equation
3544:partial difference equations
3470:Rational difference equation
2522:{\displaystyle \Delta a_{n}}
837:is a function that involves
8589:Encyclopedia of Mathematics
8332:Linear difference equations
8330:Miller, Kenneth S. (1968).
8229:R. Sedgewick, F. Flajolet,
7282:Integrodifference equations
6514:is any point on the cycle.
6012:which is less than 1.
4818:If we apply the formula to
2499:are generally omitted, and
653:. A recurrence relation of
8668:
8384:Introduction to Algorithms
8356:Brousseau, Alfred (1971).
8218:Introduction to Algorithms
8079:Lagged Fibonacci generator
8023:
5804:Matrix difference equation
5801:
5319:
3558:
3474:Matrix difference equation
2898:is defined recursively as
937:and the initial condition
605:{\displaystyle u_{0}\in X}
8561:10.1142/S0219530512500108
8437:; Robson, Robbie (2005).
8109:Infinite impulse response
7525:infinite impulse response
7521:digital signal processing
7515:Digital signal processing
2702:{\displaystyle \Delta a.}
2552:{\displaystyle \Delta a,}
1688:elements out of a set of
1548:{\displaystyle t^{2}=t+1}
1006:as its only coefficient.
8362:. Fibonacci Association.
8104:Combinatorial principles
7041:interaction is given by
2492:{\displaystyle \Delta a}
2469:{\displaystyle \Delta f}
2275:{\displaystyle \Delta ,}
1324:{\displaystyle F_{1}=1.}
972:This is an example of a
181:th term is equated to a
8418:Enders, Walter (2010).
7751:{\displaystyle \alpha }
7394:{\displaystyle c_{1}=1}
7258:the parasites, at time
6166:in the neighborhood of
5655:characteristic equation
3498:From sequences to grids
2579:applied to the element
2572:{\displaystyle \Delta }
2453:The parentheses around
2111:with the initial value
1559:of the sequence is the
1290:{\displaystyle F_{0}=0}
413:multidimensional arrays
106:that is independent of
42:according to which the
8233:, Addison-Wesley, 2013
8119:Mathematical induction
8064:Orthogonal polynomials
8007:
7878:
7752:
7732:
7718:is the output at time
7712:
7685:
7665:
7635:
7552:
7505:
7450:
7395:
7352:
7329:
7302:analysis of algorithms
7272:
7252:
7225:
7195:
7110:
7035:Nicholson–Bailey model
6987:
6861:
6703:
6676:
6535:
6508:
6478:
6415:
6395:
6372:
6307:
6306:{\displaystyle k>1}
6281:
6251:
6187:
6160:
6140:
6113:
6075:
6002:
5982:
5955:
5935:
5921:and transition matrix
5915:
5892:
5773:
5644:
5518:
5485:
5458:
5431:
5404:
5279:
5219:
5096:
5038:
4955:orthogonal polynomials
4931:
4930:{\displaystyle h\to 0}
4905:
4809:
4786:
4752:
4692:
4634:
4616:
4582:
4526:
4473:
4455:
4421:
4330:
4290:
4272:
4188:
4167:
4096:
4078:
4028:
3972:
3917:
3899:
3849:
3787:
3734:
3659:
3536:
3516:
3455:
3372:
3227:
3077:
3006:
2947:
2882:
2785:
2703:
2665:
2603:
2602:{\displaystyle a_{n}.}
2573:
2553:
2523:
2493:
2470:
2444:
2360:
2276:
2225:
2150:
2102:
1992:
1898:
1821:
1702:
1682:
1662:
1610:
1549:
1492:
1438:
1384:
1325:
1291:
1249:
1165:
1138:
1115:
1089:
997:
963:
928:
831:
781:
633:
606:
566:
520:
402:
363:
339:
323:closed-form expression
315:
288:
268:
199:
164:
140:
120:
100:
80:
56:
8606:"Recurrence Equation"
8584:"Recurrence relation"
8459:Jacques, Ian (2006).
8325:. Dover Publications.
8008:
7879:
7753:
7733:
7713:
7711:{\displaystyle y_{t}}
7686:
7671:is the input at time
7666:
7664:{\displaystyle x_{t}}
7636:
7553:
7506:
7451:
7396:
7353:
7330:
7273:
7253:
7251:{\displaystyle P_{t}}
7226:
7224:{\displaystyle N_{t}}
7196:
7111:
6988:
6862:
6704:
6677:
6570:initial value problem
6547:dyadic transformation
6536:
6509:
6507:{\displaystyle x^{*}}
6479:
6416:
6396:
6373:
6308:
6282:
6252:
6188:
6186:{\displaystyle x^{*}}
6161:
6141:
6139:{\displaystyle x^{*}}
6114:
6112:{\displaystyle x^{*}}
6076:
6003:
5983:
5981:{\displaystyle x^{*}}
5956:
5936:
5916:
5893:
5774:
5645:
5519:
5486:
5484:{\displaystyle x_{t}}
5459:
5457:{\displaystyle x_{t}}
5432:
5430:{\displaystyle w_{t}}
5405:
5280:
5220:
5097:
5039:
4932:
4906:
4810:
4766:
4726:
4666:
4635:
4596:
4556:
4500:
4474:
4435:
4395:
4304:
4291:
4252:
4189:
4141:
4097:
4058:
4002:
3952:
3918:
3879:
3829:
3767:
3735:
3660:
3537:
3517:
3456:
3373:
3254:differential equation
3228:
3078:
2986:
2948:
2891:More generally: the
2883:
2786:
2704:
2666:
2604:
2574:
2554:
2524:
2494:
2471:
2445:
2361:
2277:
2226:
2151:
2103:
1993:
1899:
1822:
1703:
1683:
1663:
1626:binomial coefficients
1620:Binomial coefficients
1611:
1550:
1493:
1439:
1385:
1326:
1292:
1250:
1166:
1164:{\displaystyle x_{0}}
1139:
1116:
1098:for a given constant
1090:
998:
964:
962:{\displaystyle 0!=1.}
929:
832:
782:
634:
632:{\displaystyle u_{0}}
607:
575:is a function, where
567:
521:
403:
364:
340:
316:
289:
269:
200:
165:
141:
121:
101:
81:
57:
8647:Recurrence relations
8527:Polyanin, Andrei D.
8517:Polyanin, Andrei D.
8371:Charles E. Leiserson
8248:Lucas, Robert E. Jr.
8026:time series analysis
7889:
7765:
7742:
7722:
7695:
7675:
7648:
7565:
7542:
7467:
7406:
7372:
7342:
7319:
7262:
7235:
7208:
7121:
7048:
7009:Mathematical biology
6877:
6716:
6693:
6578:
6525:
6491:
6428:
6405:
6385:
6320:
6291:
6271:
6204:
6170:
6150:
6123:
6096:
6027:
5992:
5965:
5945:
5925:
5905:
5815:
5663:
5531:
5508:
5468:
5441:
5414:
5330:
5235:
5116:
5054:
4968:
4915:
4822:
4645:
4484:
4301:
4204:
4112:
3930:
3745:
3675:
3577:
3526:
3506:
3388:
3301:
3093:
2960:
2902:
2798:
2723:
2687:
2618:
2583:
2563:
2537:
2503:
2480:
2457:
2383:
2293:
2263:
2160:
2115:
2008:
1919:
1834:
1830:with the base cases
1715:
1692:
1672:
1631:
1569:
1520:
1449:
1395:
1341:
1302:
1268:
1194:
1148:
1125:
1102:
1025:
987:
944:
869:
794:
667:
616:
583:
536:
450:
419:that are indexed by
392:
386:closed-form solution
353:
329:
302:
278:
213:
189:
154:
130:
110:
90:
70:
46:
18:Recurrence relations
8252:Prescott, Edward C.
8099:Time scale calculus
8054:Holonomic sequences
7033:. For example, the
7017:. For example, the
7015:population dynamics
6084:This recurrence is
5794:in absolute value.
4911:and take the limit
3492:time scale calculus
3484:Summation equations
3240:difference equation
2284:functional notation
2282:and is defined, in
2243:difference operator
1557:generating function
437:recurrence relation
36:recurrence relation
8603:Weisstein, Eric W.
8286:Sargent, Thomas J.
8216:Cormen, T. et al,
8094:Continued fraction
8003:
7874:
7748:
7728:
7708:
7681:
7661:
7631:
7548:
7501:
7446:
7391:
7348:
7325:
7310:divide and conquer
7268:
7248:
7221:
7191:
7106:
6983:
6870:by the recurrence
6857:
6699:
6672:
6531:
6504:
6474:
6411:
6391:
6368:
6303:
6277:
6247:
6183:
6156:
6146:, if the slope of
6136:
6109:
6088:, meaning that it
6071:
5998:
5978:
5951:
5931:
5911:
5901:with state vector
5891:{\displaystyle =A}
5888:
5782:The recurrence is
5769:
5640:
5514:
5481:
5454:
5427:
5400:
5398:
5275:
5215:
5092:
5034:
4927:
4901:
4805:
4630:
4469:
4286:
4184:
4092:
3913:
3730:
3655:
3532:
3512:
3488:integral equations
3451:
3368:
3223:
3073:
2943:
2878:
2781:
2699:
2661:
2599:
2569:
2549:
2519:
2489:
2466:
2440:
2356:
2272:
2221:
2146:
2098:
1988:
1894:
1886:
1859:
1817:
1698:
1678:
1658:
1656:
1606:
1545:
1488:
1434:
1380:
1321:
1287:
1260:initial conditions
1245:
1161:
1137:{\displaystyle r,}
1134:
1114:{\displaystyle r.}
1111:
1085:
993:
959:
924:
827:
777:
629:
602:
562:
516:
398:
379:holonomic function
359:
335:
314:{\displaystyle n.}
311:
284:
264:
195:
175:linear recurrences
160:
136:
116:
96:
76:
52:
8334:. W. A. Benjamin.
8220:, MIT Press, 2009
8161:978-0-415-29884-1
8059:Iterated function
7731:{\displaystyle t}
7684:{\displaystyle t}
7551:{\displaystyle T}
7463:of which will be
7351:{\displaystyle n}
7328:{\displaystyle n}
7271:{\displaystyle t}
7019:Fibonacci numbers
6853:
6806:
6803:
6759:
6756:
6702:{\displaystyle h}
6636:
6633:
6534:{\displaystyle x}
6414:{\displaystyle k}
6394:{\displaystyle f}
6280:{\displaystyle k}
6159:{\displaystyle f}
6092:to a fixed point
6001:{\displaystyle A}
5954:{\displaystyle x}
5934:{\displaystyle A}
5914:{\displaystyle x}
5517:{\displaystyle d}
5397:
5003:
4959:special functions
4798:
4628:
4538:
4467:
4284:
4179:
4090:
4040:
3984:
3911:
3861:
3799:
3535:{\displaystyle n}
3515:{\displaystyle n}
3186:
3142:
3040:
2715:second difference
2371:finite difference
2210:
2177:
2132:
2058:
2025:
1983:
1936:
1906:Pascal's triangle
1879:
1852:
1806:
1773:
1732:
1701:{\displaystyle n}
1681:{\displaystyle k}
1649:
1601:
1561:rational function
1185:linear recurrence
1181:Fibonacci numbers
1175:Fibonacci numbers
996:{\displaystyle n}
909:
762:
501:
401:{\displaystyle n}
373:functions have a
362:{\displaystyle n}
338:{\displaystyle n}
287:{\displaystyle k}
207:Fibonacci numbers
198:{\displaystyle k}
163:{\displaystyle k}
139:{\displaystyle k}
119:{\displaystyle n}
99:{\displaystyle k}
79:{\displaystyle k}
55:{\displaystyle n}
16:(Redirected from
8659:
8624:
8621:"OEIS Index Rec"
8616:
8615:
8597:
8572:
8554:
8532:
8522:
8513:
8511:
8510:
8504:
8493:
8482:
8466:
8454:
8429:
8414:
8375:Ronald L. Rivest
8367:Thomas H. Cormen
8363:
8352:
8335:
8326:
8308:
8307:
8295:
8282:Ljungqvist, Lars
8278:
8272:
8271:
8244:Stokey, Nancy L.
8240:
8234:
8227:
8221:
8214:
8208:
8207:
8201:
8193:
8191:
8190:
8184:
8177:
8169:
8163:
8150:
8144:
8141:
8012:
8010:
8009:
8004:
8002:
8001:
7983:
7982:
7970:
7969:
7951:
7950:
7929:
7928:
7901:
7900:
7883:
7881:
7880:
7875:
7870:
7869:
7845:
7844:
7805:
7804:
7777:
7776:
7757:
7755:
7754:
7749:
7737:
7735:
7734:
7729:
7717:
7715:
7714:
7709:
7707:
7706:
7690:
7688:
7687:
7682:
7670:
7668:
7667:
7662:
7660:
7659:
7640:
7638:
7637:
7632:
7627:
7626:
7605:
7604:
7577:
7576:
7557:
7555:
7554:
7549:
7510:
7508:
7507:
7502:
7485:
7484:
7455:
7453:
7452:
7447:
7445:
7444:
7440:
7418:
7417:
7400:
7398:
7397:
7392:
7384:
7383:
7357:
7355:
7354:
7349:
7334:
7332:
7331:
7326:
7296:Computer science
7277:
7275:
7274:
7269:
7257:
7255:
7254:
7249:
7247:
7246:
7230:
7228:
7227:
7222:
7220:
7219:
7200:
7198:
7197:
7192:
7184:
7183:
7182:
7181:
7152:
7151:
7139:
7138:
7115:
7113:
7112:
7107:
7105:
7104:
7103:
7102:
7082:
7081:
7066:
7065:
6992:
6990:
6989:
6984:
6973:
6972:
6960:
6959:
6944:
6943:
6931:
6930:
6909:
6908:
6896:
6895:
6866:
6864:
6863:
6858:
6851:
6835:
6834:
6816:
6815:
6804:
6801:
6788:
6787:
6769:
6768:
6757:
6754:
6747:
6746:
6728:
6727:
6708:
6706:
6705:
6700:
6689:and a step size
6681:
6679:
6678:
6673:
6668:
6667:
6652:
6651:
6634:
6631:
6588:
6561:When solving an
6540:
6538:
6537:
6532:
6513:
6511:
6510:
6505:
6503:
6502:
6483:
6481:
6480:
6475:
6464:
6456:
6455:
6443:
6435:
6420:
6418:
6417:
6412:
6400:
6398:
6397:
6392:
6377:
6375:
6374:
6369:
6312:
6310:
6309:
6304:
6286:
6284:
6283:
6278:
6256:
6254:
6253:
6248:
6240:
6232:
6231:
6219:
6211:
6193:is smaller than
6192:
6190:
6189:
6184:
6182:
6181:
6165:
6163:
6162:
6157:
6145:
6143:
6142:
6137:
6135:
6134:
6118:
6116:
6115:
6110:
6108:
6107:
6080:
6078:
6077:
6072:
6064:
6063:
6039:
6038:
6007:
6005:
6004:
5999:
5987:
5985:
5984:
5979:
5977:
5976:
5960:
5958:
5957:
5952:
5940:
5938:
5937:
5932:
5920:
5918:
5917:
5912:
5897:
5895:
5894:
5889:
5884:
5883:
5871:
5870:
5843:
5842:
5830:
5829:
5778:
5776:
5775:
5770:
5762:
5761:
5752:
5751:
5733:
5732:
5717:
5716:
5704:
5703:
5688:
5687:
5675:
5674:
5649:
5647:
5646:
5641:
5636:
5635:
5620:
5619:
5601:
5600:
5585:
5584:
5572:
5571:
5556:
5555:
5543:
5542:
5523:
5521:
5520:
5515:
5490:
5488:
5487:
5482:
5480:
5479:
5463:
5461:
5460:
5455:
5453:
5452:
5436:
5434:
5433:
5428:
5426:
5425:
5409:
5407:
5406:
5401:
5399:
5396:
5389:
5388:
5375:
5368:
5367:
5354:
5348:
5347:
5284:
5282:
5281:
5276:
5247:
5246:
5224:
5222:
5221:
5216:
5208:
5207:
5186:
5185:
5149:
5148:
5101:
5099:
5098:
5093:
5079:
5078:
5066:
5065:
5043:
5041:
5040:
5035:
5033:
5032:
5014:
5013:
5004:
4999:
4991:
4986:
4985:
4936:
4934:
4933:
4928:
4910:
4908:
4907:
4902:
4900:
4899:
4881:
4880:
4868:
4867:
4840:
4839:
4814:
4812:
4811:
4806:
4804:
4800:
4799:
4797:
4796:
4795:
4785:
4780:
4764:
4763:
4754:
4751:
4740:
4722:
4721:
4707:
4703:
4702:
4701:
4691:
4680:
4657:
4656:
4639:
4637:
4636:
4631:
4629:
4627:
4626:
4625:
4615:
4610:
4594:
4593:
4584:
4581:
4570:
4552:
4551:
4539:
4537:
4536:
4535:
4525:
4514:
4498:
4497:
4488:
4478:
4476:
4475:
4470:
4468:
4466:
4465:
4464:
4454:
4449:
4433:
4432:
4423:
4420:
4409:
4391:
4390:
4378:
4377:
4362:
4361:
4349:
4348:
4329:
4318:
4295:
4293:
4292:
4287:
4285:
4283:
4282:
4281:
4271:
4266:
4250:
4249:
4240:
4235:
4234:
4222:
4221:
4193:
4191:
4190:
4185:
4180:
4178:
4177:
4176:
4166:
4155:
4139:
4138:
4129:
4124:
4123:
4101:
4099:
4098:
4093:
4091:
4089:
4088:
4087:
4077:
4072:
4056:
4055:
4046:
4041:
4039:
4038:
4037:
4027:
4016:
4000:
3999:
3990:
3985:
3983:
3982:
3981:
3971:
3966:
3950:
3949:
3934:
3922:
3920:
3919:
3914:
3912:
3910:
3909:
3908:
3898:
3893:
3877:
3876:
3867:
3862:
3860:
3859:
3858:
3848:
3843:
3827:
3826:
3825:
3816:
3815:
3805:
3800:
3798:
3797:
3796:
3786:
3781:
3765:
3764:
3749:
3739:
3737:
3736:
3731:
3729:
3728:
3716:
3715:
3706:
3705:
3693:
3692:
3664:
3662:
3661:
3656:
3645:
3644:
3631:
3630:
3618:
3617:
3608:
3607:
3595:
3594:
3541:
3539:
3538:
3533:
3521:
3519:
3518:
3513:
3460:
3458:
3457:
3452:
3447:
3446:
3431:
3430:
3409:
3408:
3377:
3375:
3374:
3369:
3361:
3360:
3345:
3344:
3326:
3325:
3316:
3315:
3286:
3282:
3278:
3274:
3263:
3259:
3251:
3247:
3242:
3241:
3232:
3230:
3229:
3224:
3216:
3215:
3203:
3202:
3193:
3192:
3191:
3178:
3162:
3161:
3149:
3148:
3147:
3134:
3124:
3123:
3111:
3110:
3082:
3080:
3079:
3074:
3069:
3068:
3047:
3046:
3045:
3032:
3025:
3024:
3005:
3000:
2982:
2981:
2972:
2971:
2952:
2950:
2949:
2944:
2939:
2938:
2914:
2913:
2894:
2887:
2885:
2884:
2879:
2874:
2873:
2861:
2860:
2839:
2838:
2820:
2819:
2810:
2809:
2790:
2788:
2787:
2782:
2735:
2734:
2717:
2716:
2708:
2706:
2705:
2700:
2682:
2677:
2676:
2675:first difference
2670:
2668:
2667:
2662:
2657:
2656:
2655:
2639:
2638:
2608:
2606:
2605:
2600:
2595:
2594:
2578:
2576:
2575:
2570:
2558:
2556:
2555:
2550:
2533:in the sequence
2532:
2528:
2526:
2525:
2520:
2518:
2517:
2498:
2496:
2495:
2490:
2475:
2473:
2472:
2467:
2449:
2447:
2446:
2441:
2436:
2435:
2423:
2422:
2404:
2403:
2365:
2363:
2362:
2357:
2281:
2279:
2278:
2273:
2245:
2244:
2230:
2228:
2227:
2222:
2217:
2216:
2215:
2209:
2194:
2184:
2183:
2182:
2169:
2155:
2153:
2152:
2147:
2139:
2138:
2137:
2124:
2107:
2105:
2104:
2099:
2091:
2065:
2064:
2063:
2057:
2042:
2032:
2031:
2030:
2017:
1997:
1995:
1994:
1989:
1984:
1982:
1956:
1948:
1943:
1942:
1941:
1928:
1903:
1901:
1900:
1895:
1887:
1885:
1884:
1871:
1860:
1858:
1857:
1844:
1826:
1824:
1823:
1818:
1813:
1812:
1811:
1802:
1790:
1780:
1779:
1778:
1772:
1761:
1749:
1739:
1738:
1737:
1724:
1707:
1705:
1704:
1699:
1687:
1685:
1684:
1679:
1667:
1665:
1664:
1659:
1657:
1655:
1654:
1641:
1615:
1613:
1612:
1607:
1602:
1600:
1599:
1598:
1573:
1554:
1552:
1551:
1546:
1532:
1531:
1497:
1495:
1494:
1489:
1487:
1486:
1474:
1473:
1461:
1460:
1443:
1441:
1440:
1435:
1433:
1432:
1420:
1419:
1407:
1406:
1389:
1387:
1386:
1381:
1379:
1378:
1366:
1365:
1353:
1352:
1330:
1328:
1327:
1322:
1314:
1313:
1296:
1294:
1293:
1288:
1280:
1279:
1254:
1252:
1251:
1246:
1244:
1243:
1225:
1224:
1206:
1205:
1170:
1168:
1167:
1162:
1160:
1159:
1143:
1141:
1140:
1135:
1120:
1118:
1117:
1112:
1094:
1092:
1091:
1086:
1078:
1077:
1059:
1058:
1043:
1042:
1002:
1000:
999:
994:
979:
968:
966:
965:
960:
933:
931:
930:
925:
910:
907:
844:
840:
836:
834:
833:
828:
820:
819:
807:
786:
784:
783:
778:
763:
760:
754:
753:
729:
728:
710:
709:
679:
678:
659:
638:
636:
635:
630:
628:
627:
611:
609:
608:
603:
595:
594:
578:
571:
569:
568:
563:
549:
525:
523:
522:
517:
502:
499:
493:
492:
462:
461:
417:indexed families
407:
405:
404:
399:
368:
366:
365:
360:
344:
342:
341:
336:
320:
318:
317:
312:
293:
291:
290:
285:
274:where the order
273:
271:
270:
265:
263:
262:
244:
243:
225:
224:
204:
202:
201:
196:
180:
169:
167:
166:
161:
145:
143:
142:
137:
125:
123:
122:
117:
105:
103:
102:
97:
85:
83:
82:
77:
61:
59:
58:
53:
21:
8667:
8666:
8662:
8661:
8660:
8658:
8657:
8656:
8632:
8631:
8619:
8601:
8600:
8582:
8579:
8536:
8526:
8516:
8508:
8506:
8502:
8491:
8486:
8479:
8458:
8451:
8432:
8417:
8411:
8398:
8355:
8338:
8329:
8320:
8317:
8312:
8311:
8304:
8280:
8279:
8275:
8268:
8242:
8241:
8237:
8228:
8224:
8215:
8211:
8194:
8188:
8186:
8182:
8175:
8173:"Archived copy"
8171:
8170:
8166:
8151:
8147:
8142:
8138:
8133:
8128:
8123:
8049:
8032:
8022:
7984:
7974:
7955:
7942:
7920:
7892:
7887:
7886:
7852:
7830:
7796:
7768:
7763:
7762:
7740:
7739:
7720:
7719:
7698:
7693:
7692:
7673:
7672:
7651:
7646:
7645:
7612:
7596:
7568:
7563:
7562:
7540:
7539:
7529:digital filters
7517:
7476:
7465:
7464:
7461:time complexity
7428:
7409:
7404:
7403:
7375:
7370:
7369:
7340:
7339:
7317:
7316:
7298:
7260:
7259:
7238:
7233:
7232:
7211:
7206:
7205:
7173:
7162:
7143:
7124:
7119:
7118:
7094:
7083:
7073:
7051:
7046:
7045:
7011:
7006:
6964:
6951:
6935:
6922:
6900:
6881:
6875:
6874:
6826:
6807:
6779:
6760:
6738:
6719:
6714:
6713:
6691:
6690:
6659:
6643:
6581:
6576:
6575:
6559:
6523:
6522:
6494:
6489:
6488:
6447:
6436:
6426:
6425:
6403:
6402:
6383:
6382:
6318:
6317:
6289:
6288:
6269:
6268:
6223:
6212:
6202:
6201:
6173:
6168:
6167:
6148:
6147:
6126:
6121:
6120:
6099:
6094:
6093:
6049:
6030:
6025:
6024:
6018:
5990:
5989:
5968:
5963:
5962:
5943:
5942:
5923:
5922:
5903:
5902:
5875:
5856:
5834:
5821:
5813:
5812:
5806:
5800:
5753:
5743:
5718:
5708:
5689:
5679:
5666:
5661:
5660:
5621:
5611:
5586:
5576:
5557:
5547:
5534:
5529:
5528:
5506:
5505:
5502:
5497:
5471:
5466:
5465:
5444:
5439:
5438:
5417:
5412:
5411:
5380:
5376:
5359:
5355:
5333:
5328:
5327:
5324:
5318:
5238:
5233:
5232:
5193:
5177:
5134:
5114:
5113:
5107:Bessel function
5070:
5057:
5052:
5051:
5018:
5005:
4992:
4971:
4966:
4965:
4947:
4913:
4912:
4888:
4872:
4856:
4825:
4820:
4819:
4787:
4765:
4755:
4713:
4712:
4708:
4693:
4665:
4661:
4648:
4643:
4642:
4617:
4595:
4585:
4543:
4527:
4499:
4489:
4482:
4481:
4456:
4434:
4424:
4382:
4369:
4353:
4334:
4299:
4298:
4273:
4251:
4241:
4226:
4207:
4202:
4201:
4168:
4140:
4130:
4115:
4110:
4109:
4079:
4057:
4047:
4029:
4001:
3991:
3973:
3951:
3935:
3928:
3927:
3900:
3878:
3868:
3850:
3828:
3817:
3807:
3806:
3788:
3766:
3750:
3743:
3742:
3720:
3707:
3697:
3678:
3673:
3672:
3636:
3622:
3609:
3599:
3580:
3575:
3574:
3568:
3563:
3557:
3552:
3524:
3523:
3504:
3503:
3500:
3438:
3416:
3394:
3386:
3385:
3352:
3336:
3317:
3307:
3299:
3298:
3284:
3280:
3276:
3272:
3268:of a function.
3261:
3257:
3249:
3245:
3239:
3238:
3207:
3194:
3173:
3153:
3129:
3115:
3096:
3091:
3090:
3048:
3027:
3016:
2973:
2963:
2958:
2957:
2924:
2905:
2900:
2899:
2892:
2865:
2846:
2824:
2811:
2801:
2796:
2795:
2726:
2721:
2720:
2714:
2713:
2685:
2684:
2680:
2674:
2673:
2640:
2630:
2616:
2615:
2586:
2581:
2580:
2561:
2560:
2535:
2534:
2530:
2509:
2501:
2500:
2478:
2477:
2455:
2454:
2427:
2408:
2395:
2381:
2380:
2291:
2290:
2261:
2260:
2242:
2241:
2237:
2199:
2189:
2164:
2158:
2157:
2119:
2113:
2112:
2047:
2037:
2012:
2006:
2005:
1957:
1949:
1923:
1917:
1916:
1866:
1839:
1832:
1831:
1792:
1785:
1762:
1751:
1744:
1719:
1713:
1712:
1690:
1689:
1670:
1669:
1636:
1629:
1628:
1622:
1590:
1577:
1567:
1566:
1523:
1518:
1517:
1514:Binet's formula
1478:
1465:
1452:
1447:
1446:
1424:
1411:
1398:
1393:
1392:
1370:
1357:
1344:
1339:
1338:
1305:
1300:
1299:
1271:
1266:
1265:
1229:
1210:
1197:
1192:
1191:
1177:
1151:
1146:
1145:
1123:
1122:
1100:
1099:
1069:
1050:
1028:
1023:
1022:
1012:
985:
984:
977:
942:
941:
867:
866:
856:
851:
842:
838:
811:
792:
791:
739:
714:
695:
670:
665:
664:
657:
619:
614:
613:
586:
581:
580:
576:
534:
533:
478:
453:
448:
447:
433:
425:natural numbers
390:
389:
351:
350:
327:
326:
300:
299:
276:
275:
248:
229:
216:
211:
210:
187:
186:
183:linear function
178:
152:
151:
128:
127:
108:
107:
88:
87:
68:
67:
44:
43:
28:
23:
22:
15:
12:
11:
5:
8665:
8663:
8655:
8654:
8649:
8644:
8634:
8633:
8630:
8629:
8617:
8598:
8578:
8577:External links
8575:
8574:
8573:
8545:(2): 215–235.
8534:
8524:
8514:
8484:
8477:
8456:
8449:
8430:
8428:on 2014-11-10.
8415:
8409:
8396:
8379:Clifford Stein
8364:
8353:
8336:
8327:
8316:
8313:
8310:
8309:
8302:
8273:
8266:
8235:
8222:
8209:
8164:
8145:
8135:
8134:
8132:
8129:
8127:
8124:
8122:
8121:
8116:
8111:
8106:
8101:
8096:
8091:
8086:
8081:
8076:
8071:
8066:
8061:
8056:
8050:
8048:
8045:
8021:
8018:
8014:
8013:
8000:
7997:
7994:
7991:
7987:
7981:
7977:
7973:
7968:
7965:
7962:
7958:
7954:
7949:
7945:
7941:
7938:
7935:
7932:
7927:
7923:
7919:
7916:
7913:
7910:
7907:
7904:
7899:
7895:
7884:
7873:
7868:
7865:
7862:
7859:
7855:
7851:
7848:
7843:
7840:
7837:
7833:
7829:
7826:
7823:
7820:
7817:
7814:
7811:
7808:
7803:
7799:
7795:
7792:
7789:
7786:
7783:
7780:
7775:
7771:
7747:
7727:
7705:
7701:
7680:
7658:
7654:
7642:
7641:
7630:
7625:
7622:
7619:
7615:
7611:
7608:
7603:
7599:
7595:
7592:
7589:
7586:
7583:
7580:
7575:
7571:
7547:
7516:
7513:
7500:
7497:
7494:
7491:
7488:
7483:
7479:
7475:
7472:
7457:
7456:
7443:
7439:
7435:
7431:
7427:
7424:
7421:
7416:
7412:
7401:
7390:
7387:
7382:
7378:
7347:
7324:
7297:
7294:
7267:
7245:
7241:
7218:
7214:
7202:
7201:
7190:
7187:
7180:
7176:
7172:
7169:
7165:
7161:
7158:
7155:
7150:
7146:
7142:
7137:
7134:
7131:
7127:
7116:
7101:
7097:
7093:
7090:
7086:
7080:
7076:
7072:
7069:
7064:
7061:
7058:
7054:
7010:
7007:
7005:
7002:
6998:discretization
6994:
6993:
6982:
6979:
6976:
6971:
6967:
6963:
6958:
6954:
6950:
6947:
6942:
6938:
6934:
6929:
6925:
6921:
6918:
6915:
6912:
6907:
6903:
6899:
6894:
6891:
6888:
6884:
6868:
6867:
6856:
6850:
6847:
6844:
6841:
6838:
6833:
6829:
6825:
6822:
6819:
6814:
6810:
6800:
6797:
6794:
6791:
6786:
6782:
6778:
6775:
6772:
6767:
6763:
6753:
6750:
6745:
6741:
6737:
6734:
6731:
6726:
6722:
6698:
6687:Euler's method
6683:
6682:
6671:
6666:
6662:
6658:
6655:
6650:
6646:
6642:
6639:
6630:
6627:
6624:
6621:
6618:
6615:
6612:
6609:
6606:
6603:
6600:
6597:
6594:
6591:
6587:
6584:
6558:
6555:
6530:
6501:
6497:
6485:
6484:
6473:
6470:
6467:
6463:
6459:
6454:
6450:
6446:
6442:
6439:
6434:
6410:
6390:
6379:
6378:
6367:
6364:
6361:
6358:
6355:
6352:
6349:
6346:
6343:
6340:
6337:
6334:
6331:
6328:
6325:
6302:
6299:
6296:
6276:
6258:
6257:
6246:
6243:
6239:
6235:
6230:
6226:
6222:
6218:
6215:
6210:
6180:
6176:
6155:
6133:
6129:
6106:
6102:
6086:locally stable
6082:
6081:
6070:
6067:
6062:
6059:
6056:
6052:
6048:
6045:
6042:
6037:
6033:
6017:
6014:
6010:absolute value
5997:
5975:
5971:
5950:
5930:
5910:
5899:
5898:
5887:
5882:
5878:
5874:
5869:
5866:
5863:
5859:
5855:
5852:
5849:
5846:
5841:
5837:
5833:
5828:
5824:
5820:
5802:Main article:
5799:
5796:
5780:
5779:
5768:
5765:
5760:
5756:
5750:
5746:
5742:
5739:
5736:
5731:
5728:
5725:
5721:
5715:
5711:
5707:
5702:
5699:
5696:
5692:
5686:
5682:
5678:
5673:
5669:
5651:
5650:
5639:
5634:
5631:
5628:
5624:
5618:
5614:
5610:
5607:
5604:
5599:
5596:
5593:
5589:
5583:
5579:
5575:
5570:
5567:
5564:
5560:
5554:
5550:
5546:
5541:
5537:
5513:
5501:
5498:
5496:
5493:
5478:
5474:
5451:
5447:
5424:
5420:
5395:
5392:
5387:
5383:
5379:
5374:
5371:
5366:
5362:
5358:
5351:
5346:
5343:
5340:
5336:
5320:Main article:
5317:
5314:
5310:hypergeometric
5286:
5285:
5274:
5271:
5268:
5265:
5262:
5259:
5256:
5253:
5250:
5245:
5241:
5226:
5225:
5214:
5211:
5206:
5203:
5200:
5196:
5192:
5189:
5184:
5180:
5176:
5173:
5170:
5167:
5164:
5161:
5158:
5155:
5152:
5147:
5144:
5141:
5137:
5133:
5130:
5127:
5124:
5121:
5103:
5102:
5091:
5088:
5085:
5082:
5077:
5073:
5069:
5064:
5060:
5045:
5044:
5031:
5028:
5025:
5021:
5017:
5012:
5008:
5002:
4998:
4995:
4989:
4984:
4981:
4978:
4974:
4946:
4943:
4926:
4923:
4920:
4898:
4895:
4891:
4887:
4884:
4879:
4875:
4871:
4866:
4863:
4859:
4855:
4852:
4849:
4846:
4843:
4838:
4835:
4832:
4828:
4816:
4815:
4803:
4794:
4790:
4784:
4779:
4776:
4773:
4769:
4762:
4758:
4750:
4747:
4744:
4739:
4736:
4733:
4729:
4725:
4720:
4716:
4711:
4706:
4700:
4696:
4690:
4687:
4684:
4679:
4676:
4673:
4669:
4664:
4660:
4655:
4651:
4640:
4624:
4620:
4614:
4609:
4606:
4603:
4599:
4592:
4588:
4580:
4577:
4574:
4569:
4566:
4563:
4559:
4555:
4550:
4546:
4542:
4534:
4530:
4524:
4521:
4518:
4513:
4510:
4507:
4503:
4496:
4492:
4479:
4463:
4459:
4453:
4448:
4445:
4442:
4438:
4431:
4427:
4419:
4416:
4413:
4408:
4405:
4402:
4398:
4394:
4389:
4385:
4381:
4376:
4372:
4368:
4365:
4360:
4356:
4352:
4347:
4344:
4341:
4337:
4333:
4328:
4325:
4322:
4317:
4314:
4311:
4307:
4296:
4280:
4276:
4270:
4265:
4262:
4259:
4255:
4248:
4244:
4238:
4233:
4229:
4225:
4220:
4217:
4214:
4210:
4195:
4194:
4183:
4175:
4171:
4165:
4162:
4159:
4154:
4151:
4148:
4144:
4137:
4133:
4127:
4122:
4118:
4103:
4102:
4086:
4082:
4076:
4071:
4068:
4065:
4061:
4054:
4050:
4044:
4036:
4032:
4026:
4023:
4020:
4015:
4012:
4009:
4005:
3998:
3994:
3988:
3980:
3976:
3970:
3965:
3962:
3959:
3955:
3948:
3945:
3942:
3938:
3924:
3923:
3907:
3903:
3897:
3892:
3889:
3886:
3882:
3875:
3871:
3865:
3857:
3853:
3847:
3842:
3839:
3836:
3832:
3824:
3820:
3814:
3810:
3803:
3795:
3791:
3785:
3780:
3777:
3774:
3770:
3763:
3760:
3757:
3753:
3740:
3727:
3723:
3719:
3714:
3710:
3704:
3700:
3696:
3691:
3688:
3685:
3681:
3666:
3665:
3654:
3651:
3648:
3643:
3639:
3634:
3629:
3625:
3621:
3616:
3612:
3606:
3602:
3598:
3593:
3590:
3587:
3583:
3567:
3564:
3559:Main article:
3556:
3553:
3551:
3548:
3531:
3511:
3499:
3496:
3462:
3461:
3450:
3445:
3441:
3437:
3434:
3429:
3426:
3423:
3419:
3415:
3412:
3407:
3404:
3401:
3397:
3393:
3379:
3378:
3367:
3364:
3359:
3355:
3351:
3348:
3343:
3339:
3335:
3332:
3329:
3324:
3320:
3314:
3310:
3306:
3234:
3233:
3222:
3219:
3214:
3210:
3206:
3201:
3197:
3190:
3185:
3182:
3177:
3171:
3168:
3165:
3160:
3156:
3152:
3146:
3141:
3138:
3133:
3127:
3122:
3118:
3114:
3109:
3106:
3103:
3099:
3084:
3083:
3072:
3067:
3064:
3061:
3058:
3055:
3051:
3044:
3039:
3036:
3031:
3023:
3019:
3015:
3012:
3009:
3004:
2999:
2996:
2993:
2989:
2985:
2980:
2976:
2970:
2966:
2942:
2937:
2934:
2931:
2927:
2923:
2920:
2917:
2912:
2908:
2889:
2888:
2877:
2872:
2868:
2864:
2859:
2856:
2853:
2849:
2845:
2842:
2837:
2834:
2831:
2827:
2823:
2818:
2814:
2808:
2804:
2780:
2777:
2774:
2771:
2768:
2765:
2762:
2759:
2756:
2753:
2750:
2747:
2744:
2741:
2738:
2733:
2729:
2698:
2695:
2692:
2660:
2654:
2650:
2647:
2643:
2637:
2633:
2629:
2626:
2623:
2598:
2593:
2589:
2568:
2548:
2545:
2542:
2516:
2512:
2508:
2488:
2485:
2465:
2462:
2451:
2450:
2439:
2434:
2430:
2426:
2421:
2418:
2415:
2411:
2407:
2402:
2398:
2394:
2391:
2388:
2367:
2366:
2355:
2352:
2349:
2346:
2343:
2340:
2337:
2334:
2331:
2328:
2325:
2322:
2319:
2316:
2313:
2310:
2307:
2304:
2301:
2298:
2271:
2268:
2236:
2233:
2220:
2214:
2208:
2205:
2202:
2198:
2193:
2187:
2181:
2176:
2173:
2168:
2145:
2142:
2136:
2131:
2128:
2123:
2109:
2108:
2097:
2094:
2090:
2086:
2083:
2080:
2077:
2074:
2071:
2068:
2062:
2056:
2053:
2050:
2046:
2041:
2035:
2029:
2024:
2021:
2016:
1999:
1998:
1987:
1981:
1978:
1975:
1972:
1969:
1966:
1963:
1960:
1955:
1952:
1946:
1940:
1935:
1932:
1927:
1893:
1890:
1883:
1878:
1875:
1870:
1863:
1856:
1851:
1848:
1843:
1828:
1827:
1816:
1810:
1805:
1801:
1798:
1795:
1789:
1783:
1777:
1771:
1768:
1765:
1760:
1757:
1754:
1748:
1742:
1736:
1731:
1728:
1723:
1697:
1677:
1653:
1648:
1645:
1640:
1621:
1618:
1617:
1616:
1605:
1597:
1593:
1589:
1586:
1583:
1580:
1576:
1544:
1541:
1538:
1535:
1530:
1526:
1510:
1509:
1499:
1498:
1485:
1481:
1477:
1472:
1468:
1464:
1459:
1455:
1444:
1431:
1427:
1423:
1418:
1414:
1410:
1405:
1401:
1390:
1377:
1373:
1369:
1364:
1360:
1356:
1351:
1347:
1332:
1331:
1320:
1317:
1312:
1308:
1297:
1286:
1283:
1278:
1274:
1256:
1255:
1242:
1239:
1236:
1232:
1228:
1223:
1220:
1217:
1213:
1209:
1204:
1200:
1176:
1173:
1158:
1154:
1133:
1130:
1110:
1107:
1096:
1095:
1084:
1081:
1076:
1072:
1068:
1065:
1062:
1057:
1053:
1049:
1046:
1041:
1038:
1035:
1031:
1011:
1008:
1004:
1003:
992:
970:
969:
958:
955:
952:
949:
935:
934:
923:
920:
917:
914:
904:
901:
898:
895:
892:
889:
886:
883:
880:
877:
874:
855:
852:
850:
847:
826:
823:
818:
814:
810:
806:
802:
799:
788:
787:
776:
773:
770:
767:
757:
752:
749:
746:
742:
738:
735:
732:
727:
724:
721:
717:
713:
708:
705:
702:
698:
694:
691:
688:
685:
682:
677:
673:
660:has the form
626:
622:
601:
598:
593:
589:
573:
572:
561:
558:
555:
552:
548:
544:
541:
527:
526:
515:
512:
509:
506:
496:
491:
488:
485:
481:
477:
474:
471:
468:
465:
460:
456:
432:
429:
397:
358:
334:
310:
307:
283:
261:
258:
255:
251:
247:
242:
239:
236:
232:
228:
223:
219:
194:
159:
146:is called the
135:
126:; this number
115:
95:
75:
51:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8664:
8653:
8652:Combinatorics
8650:
8648:
8645:
8643:
8640:
8639:
8637:
8627:
8622:
8618:
8613:
8612:
8607:
8604:
8599:
8595:
8591:
8590:
8585:
8581:
8580:
8576:
8570:
8566:
8562:
8558:
8553:
8548:
8544:
8540:
8535:
8530:
8525:
8520:
8515:
8505:on 2016-03-04
8501:
8497:
8490:
8485:
8480:
8478:0-273-70195-9
8474:
8470:
8465:
8464:
8457:
8452:
8450:0-387-23234-6
8446:
8442:
8441:
8436:
8435:Flahive, Mary
8431:
8427:
8423:
8422:
8416:
8412:
8410:0-201-55802-5
8406:
8402:
8397:
8394:
8393:0-262-03293-7
8390:
8386:
8385:
8380:
8376:
8372:
8368:
8365:
8361:
8360:
8354:
8350:
8346:
8342:
8337:
8333:
8328:
8324:
8319:
8318:
8314:
8305:
8303:0-262-12274-X
8299:
8294:
8293:
8287:
8283:
8277:
8274:
8269:
8267:0-674-75096-9
8263:
8259:
8258:
8253:
8249:
8245:
8239:
8236:
8232:
8226:
8223:
8219:
8213:
8210:
8205:
8199:
8181:
8174:
8168:
8165:
8162:
8158:
8154:
8149:
8146:
8140:
8137:
8130:
8125:
8120:
8117:
8115:
8112:
8110:
8107:
8105:
8102:
8100:
8097:
8095:
8092:
8090:
8087:
8085:
8082:
8080:
8077:
8075:
8072:
8070:
8067:
8065:
8062:
8060:
8057:
8055:
8052:
8051:
8046:
8044:
8042:
8038:
8037:interest rate
8031:
8027:
8019:
8017:
7998:
7995:
7992:
7989:
7985:
7979:
7975:
7971:
7966:
7963:
7960:
7956:
7947:
7943:
7939:
7936:
7930:
7925:
7921:
7914:
7911:
7908:
7902:
7897:
7893:
7885:
7866:
7863:
7860:
7857:
7853:
7849:
7846:
7841:
7838:
7835:
7831:
7824:
7821:
7818:
7809:
7806:
7801:
7797:
7790:
7787:
7784:
7778:
7773:
7769:
7761:
7760:
7759:
7745:
7725:
7703:
7699:
7678:
7656:
7652:
7628:
7623:
7620:
7617:
7613:
7609:
7606:
7601:
7597:
7590:
7587:
7584:
7578:
7573:
7569:
7561:
7560:
7559:
7545:
7537:
7532:
7530:
7526:
7522:
7514:
7512:
7492:
7486:
7481:
7477:
7470:
7462:
7441:
7437:
7433:
7429:
7425:
7422:
7419:
7414:
7410:
7402:
7388:
7385:
7380:
7376:
7368:
7367:
7366:
7364:
7363:binary search
7359:
7345:
7336:
7322:
7313:
7311:
7307:
7303:
7295:
7293:
7292:populations.
7291:
7287:
7283:
7279:
7265:
7243:
7239:
7216:
7212:
7188:
7178:
7174:
7170:
7167:
7163:
7159:
7156:
7148:
7144:
7140:
7135:
7132:
7129:
7125:
7117:
7099:
7095:
7091:
7088:
7084:
7078:
7074:
7070:
7067:
7062:
7059:
7056:
7052:
7044:
7043:
7042:
7040:
7036:
7032:
7027:
7022:
7020:
7016:
7008:
7003:
7001:
6999:
6980:
6977:
6974:
6969:
6965:
6961:
6956:
6952:
6948:
6940:
6936:
6932:
6927:
6923:
6916:
6913:
6910:
6905:
6901:
6897:
6892:
6889:
6886:
6882:
6873:
6872:
6871:
6854:
6848:
6842:
6839:
6836:
6831:
6827:
6820:
6817:
6812:
6808:
6798:
6792:
6789:
6784:
6780:
6773:
6770:
6765:
6761:
6751:
6743:
6739:
6732:
6729:
6724:
6720:
6712:
6711:
6710:
6696:
6688:
6669:
6664:
6660:
6656:
6648:
6644:
6637:
6628:
6619:
6613:
6610:
6607:
6601:
6598:
6592:
6585:
6582:
6574:
6573:
6572:
6571:
6567:
6564:
6556:
6554:
6552:
6548:
6544:
6528:
6520:
6515:
6499:
6495:
6471:
6468:
6465:
6452:
6448:
6440:
6437:
6424:
6423:
6422:
6408:
6388:
6362:
6356:
6353:
6350:
6347:
6344:
6341:
6338:
6335:
6329:
6323:
6316:
6315:
6314:
6300:
6297:
6294:
6274:
6265:
6263:
6244:
6241:
6228:
6224:
6216:
6213:
6200:
6199:
6198:
6196:
6178:
6174:
6153:
6131:
6127:
6104:
6100:
6091:
6087:
6068:
6060:
6057:
6054:
6050:
6043:
6040:
6035:
6031:
6023:
6022:
6021:
6015:
6013:
6011:
5995:
5973:
5969:
5948:
5928:
5908:
5880:
5876:
5872:
5867:
5864:
5861:
5857:
5850:
5847:
5839:
5835:
5831:
5826:
5822:
5811:
5810:
5809:
5805:
5797:
5795:
5793:
5789:
5785:
5766:
5763:
5758:
5754:
5748:
5744:
5740:
5737:
5734:
5729:
5726:
5723:
5719:
5713:
5709:
5705:
5700:
5697:
5694:
5690:
5684:
5680:
5676:
5671:
5667:
5659:
5658:
5657:
5656:
5637:
5632:
5629:
5626:
5622:
5616:
5612:
5608:
5605:
5602:
5597:
5594:
5591:
5587:
5581:
5577:
5573:
5568:
5565:
5562:
5558:
5552:
5548:
5544:
5539:
5535:
5527:
5526:
5525:
5511:
5499:
5494:
5492:
5476:
5472:
5449:
5445:
5422:
5418:
5393:
5390:
5385:
5381:
5377:
5372:
5369:
5364:
5360:
5356:
5349:
5344:
5341:
5338:
5334:
5323:
5315:
5313:
5311:
5307:
5303:
5299:
5295:
5291:
5269:
5266:
5263:
5260:
5257:
5251:
5248:
5243:
5239:
5231:
5230:
5229:
5228:is solved by
5212:
5209:
5204:
5201:
5198:
5194:
5190:
5187:
5182:
5178:
5171:
5168:
5165:
5162:
5159:
5156:
5150:
5145:
5142:
5139:
5135:
5128:
5125:
5122:
5112:
5111:
5110:
5108:
5089:
5083:
5075:
5071:
5067:
5062:
5058:
5050:
5049:
5048:
5029:
5026:
5023:
5019:
5015:
5010:
5006:
5000:
4996:
4993:
4987:
4982:
4979:
4976:
4972:
4964:
4963:
4962:
4960:
4956:
4952:
4944:
4942:
4940:
4924:
4918:
4896:
4893:
4889:
4885:
4882:
4877:
4873:
4864:
4861:
4857:
4853:
4850:
4847:
4841:
4836:
4833:
4830:
4826:
4801:
4792:
4788:
4782:
4777:
4774:
4771:
4767:
4760:
4756:
4748:
4745:
4742:
4737:
4734:
4731:
4727:
4723:
4718:
4714:
4709:
4704:
4698:
4694:
4688:
4685:
4682:
4677:
4674:
4671:
4667:
4662:
4658:
4653:
4649:
4641:
4622:
4618:
4612:
4607:
4604:
4601:
4597:
4590:
4586:
4578:
4575:
4572:
4567:
4564:
4561:
4557:
4553:
4548:
4544:
4540:
4532:
4528:
4522:
4519:
4516:
4511:
4508:
4505:
4501:
4494:
4490:
4480:
4461:
4457:
4451:
4446:
4443:
4440:
4436:
4429:
4425:
4417:
4414:
4411:
4406:
4403:
4400:
4396:
4392:
4387:
4383:
4379:
4374:
4370:
4366:
4358:
4354:
4350:
4345:
4342:
4339:
4335:
4326:
4323:
4320:
4315:
4312:
4309:
4305:
4297:
4278:
4274:
4268:
4263:
4260:
4257:
4253:
4246:
4242:
4236:
4231:
4227:
4223:
4218:
4215:
4212:
4208:
4200:
4199:
4198:
4181:
4173:
4169:
4163:
4160:
4157:
4152:
4149:
4146:
4142:
4135:
4131:
4125:
4120:
4116:
4108:
4107:
4106:
4084:
4080:
4074:
4069:
4066:
4063:
4059:
4052:
4048:
4042:
4034:
4030:
4024:
4021:
4018:
4013:
4010:
4007:
4003:
3996:
3992:
3986:
3978:
3974:
3968:
3963:
3960:
3957:
3953:
3946:
3943:
3940:
3936:
3926:
3925:
3905:
3901:
3895:
3890:
3887:
3884:
3880:
3873:
3869:
3863:
3855:
3851:
3845:
3840:
3837:
3834:
3830:
3822:
3818:
3812:
3808:
3801:
3793:
3789:
3783:
3778:
3775:
3772:
3768:
3761:
3758:
3755:
3751:
3741:
3725:
3721:
3717:
3712:
3708:
3702:
3698:
3694:
3689:
3686:
3683:
3679:
3671:
3670:
3669:
3652:
3649:
3646:
3641:
3637:
3632:
3627:
3623:
3619:
3614:
3610:
3604:
3600:
3596:
3591:
3588:
3585:
3581:
3573:
3572:
3571:
3565:
3562:
3554:
3549:
3547:
3545:
3529:
3509:
3497:
3495:
3493:
3489:
3485:
3481:
3477:
3475:
3471:
3465:
3448:
3443:
3439:
3435:
3432:
3427:
3424:
3421:
3417:
3413:
3410:
3405:
3402:
3399:
3395:
3391:
3384:
3383:
3382:
3365:
3362:
3357:
3353:
3349:
3346:
3341:
3337:
3330:
3327:
3322:
3318:
3312:
3304:
3297:
3296:
3295:
3292:
3290:
3269:
3267:
3255:
3243:
3220:
3212:
3208:
3199:
3183:
3180:
3169:
3166:
3163:
3158:
3154:
3139:
3136:
3125:
3120:
3116:
3112:
3107:
3104:
3101:
3097:
3089:
3088:
3087:
3070:
3065:
3062:
3059:
3056:
3053:
3049:
3037:
3034:
3021:
3013:
3010:
3002:
2997:
2994:
2991:
2987:
2983:
2978:
2974:
2968:
2956:
2955:
2954:
2940:
2935:
2932:
2929:
2921:
2915:
2910:
2897:
2896:th difference
2875:
2870:
2866:
2862:
2857:
2854:
2851:
2847:
2843:
2840:
2835:
2832:
2829:
2825:
2821:
2816:
2812:
2806:
2794:
2793:
2792:
2778:
2772:
2760:
2757:
2748:
2739:
2736:
2731:
2718:
2709:
2696:
2693:
2678:
2658:
2648:
2645:
2635:
2631:
2624:
2621:
2614:
2609:
2596:
2591:
2587:
2546:
2543:
2514:
2510:
2486:
2463:
2437:
2432:
2428:
2424:
2419:
2416:
2413:
2409:
2405:
2400:
2392:
2379:
2378:
2377:
2374:
2372:
2353:
2347:
2341:
2338:
2332:
2329:
2326:
2320:
2317:
2311:
2302:
2289:
2288:
2287:
2285:
2269:
2258:
2254:
2250:
2246:
2234:
2232:
2218:
2206:
2203:
2200:
2196:
2185:
2174:
2171:
2143:
2140:
2129:
2126:
2095:
2092:
2088:
2081:
2078:
2075:
2072:
2069:
2054:
2051:
2048:
2044:
2033:
2022:
2019:
2004:
2003:
2002:
1985:
1979:
1973:
1970:
1967:
1961:
1958:
1953:
1950:
1944:
1933:
1930:
1915:
1914:
1913:
1911:
1907:
1891:
1888:
1876:
1873:
1861:
1849:
1846:
1814:
1803:
1799:
1796:
1793:
1781:
1769:
1766:
1763:
1758:
1755:
1752:
1740:
1729:
1726:
1711:
1710:
1709:
1695:
1675:
1646:
1643:
1627:
1619:
1603:
1595:
1591:
1587:
1584:
1581:
1578:
1574:
1565:
1564:
1563:
1562:
1558:
1542:
1539:
1536:
1533:
1528:
1524:
1515:
1507:
1506:
1505:
1502:
1483:
1479:
1475:
1470:
1466:
1462:
1457:
1453:
1445:
1429:
1425:
1421:
1416:
1412:
1408:
1403:
1399:
1391:
1375:
1371:
1367:
1362:
1358:
1354:
1349:
1345:
1337:
1336:
1335:
1318:
1315:
1310:
1306:
1298:
1284:
1281:
1276:
1272:
1264:
1263:
1262:
1261:
1240:
1237:
1234:
1230:
1226:
1221:
1218:
1215:
1211:
1207:
1202:
1198:
1190:
1189:
1188:
1186:
1182:
1174:
1172:
1156:
1152:
1131:
1128:
1108:
1105:
1082:
1074:
1070:
1066:
1063:
1055:
1051:
1047:
1044:
1039:
1036:
1033:
1029:
1021:
1020:
1019:
1017:
1009:
1007:
990:
983:
982:
981:
975:
956:
953:
950:
947:
940:
939:
938:
921:
918:
915:
912:
902:
896:
893:
890:
884:
881:
878:
875:
872:
865:
864:
863:
861:
853:
848:
846:
824:
816:
812:
808:
800:
797:
774:
771:
768:
765:
750:
747:
744:
740:
736:
733:
730:
725:
722:
719:
715:
711:
706:
703:
700:
696:
692:
689:
683:
680:
675:
671:
663:
662:
661:
656:
652:
647:
644:
642:
641:initial value
624:
620:
599:
596:
591:
587:
559:
553:
550:
542:
539:
532:
531:
530:
513:
510:
507:
504:
489:
486:
483:
479:
475:
472:
466:
463:
458:
454:
446:
445:
444:
442:
438:
430:
428:
426:
422:
418:
414:
409:
395:
387:
382:
380:
376:
375:Taylor series
372:
356:
349:depending on
348:
332:
324:
308:
305:
297:
281:
259:
256:
253:
249:
245:
240:
237:
234:
230:
226:
221:
217:
208:
192:
184:
176:
171:
157:
149:
133:
113:
93:
73:
65:
62:th term of a
49:
41:
37:
33:
19:
8609:
8587:
8542:
8538:
8507:. Retrieved
8500:the original
8495:
8462:
8443:. Springer.
8439:
8433:Cull, Paul;
8426:the original
8420:
8400:
8382:
8358:
8340:
8331:
8322:
8315:Bibliography
8291:
8276:
8256:
8238:
8230:
8225:
8217:
8212:
8187:. Retrieved
8167:
8148:
8139:
8033:
8015:
7643:
7533:
7518:
7458:
7360:
7337:
7314:
7299:
7280:
7203:
7026:logistic map
7023:
7012:
7004:Applications
6995:
6869:
6684:
6560:
6543:logistic map
6516:
6486:
6380:
6266:
6261:
6259:
6083:
6019:
5900:
5807:
5781:
5652:
5503:
5325:
5287:
5227:
5104:
5047:is given by
5046:
4948:
4817:
4196:
4104:
3667:
3569:
3543:
3501:
3482:
3478:
3466:
3463:
3380:
3293:
3270:
3260:relates the
3237:
3235:
3085:
2953:and one has
2895:
2890:
2712:
2710:
2672:
2610:
2452:
2375:
2368:
2240:
2238:
2110:
2000:
1829:
1623:
1511:
1503:
1500:
1333:
1257:
1178:
1097:
1016:logistic map
1013:
1010:Logistic map
1005:
973:
971:
936:
857:
789:
654:
650:
648:
645:
640:
574:
528:
436:
434:
410:
383:
174:
172:
147:
35:
29:
7536:comb filter
7037:for a host-
7031:populations
6566:numerically
5788:eigenvalues
5312:solutions.
5298:P-recursive
5296:are called
4957:, and many
3266:derivatives
1018:defined by
651:first order
415:, that is,
345:. As well,
32:mathematics
8636:Categories
8539:Anal. Appl
8509:2014-08-07
8455:chapter 7.
8189:2010-10-19
8126:References
8024:See also:
7290:univoltine
6401:appearing
5302:polynomial
2251:that maps
1910:factorials
431:Definition
8611:MathWorld
8594:EMS Press
8552:1101.4371
8131:Footnotes
8069:Recursion
8020:Economics
7993:−
7976:α
7964:−
7944:α
7940:−
7937:α
7915:α
7912:−
7861:−
7850:α
7839:−
7825:α
7822:−
7810:α
7791:α
7788:−
7746:α
7621:−
7610:α
7591:α
7588:−
7538:of delay
7487:
7306:algorithm
7168:−
7160:−
7089:−
7071:λ
7000:article.
6855:…
6500:∗
6453:∗
6354:∘
6351:⋯
6348:∘
6342:∘
6229:∗
6179:∗
6132:∗
6105:∗
6090:converges
6058:−
5974:∗
5881:∗
5873:−
5865:−
5840:∗
5832:−
5755:λ
5741:−
5738:⋯
5735:−
5727:−
5720:λ
5706:−
5698:−
5691:λ
5677:−
5668:λ
5630:−
5606:⋯
5595:−
5566:−
5495:Stability
5188:−
5163:−
5143:−
5126:−
5027:−
5016:−
4922:→
4768:∏
4746:−
4728:∑
4686:−
4668:∏
4598:∏
4576:−
4558:∑
4520:−
4502:∏
4437:∏
4415:−
4397:∑
4380:−
4351:−
4324:−
4306:∑
4254:∏
4224:−
4161:−
4143:∏
4060:∏
4022:−
4004:∏
3987:−
3954:∏
3881:∏
3831:∏
3802:−
3769:∏
3695:−
3647:≠
3433:−
3334:Δ
3309:Δ
3256:of order
3244:of order
3196:Δ
3167:⋯
3151:Δ
3063:−
3011:−
2988:∑
2965:Δ
2933:−
2926:Δ
2922:∘
2919:Δ
2907:Δ
2841:−
2803:Δ
2770:Δ
2764:Δ
2752:Δ
2749:∘
2746:Δ
2728:Δ
2691:Δ
2649:∈
2567:Δ
2541:Δ
2507:Δ
2484:Δ
2461:Δ
2425:−
2390:Δ
2339:−
2300:Δ
2267:Δ
2257:functions
2253:sequences
2204:−
2073:−
2052:−
1971:−
1797:−
1767:−
1756:−
1588:−
1582:−
1238:−
1219:−
1067:−
894:−
885:⋅
860:factorial
854:Factorial
822:→
809:×
798:φ
769:≥
748:−
734:…
723:−
704:−
684:φ
597:∈
557:→
551:×
540:φ
487:−
467:φ
257:−
238:−
8569:28828175
8341:SIAM Rev
8288:(2004).
8254:(1989).
8198:cite web
8180:Archived
8047:See also
7304:. If an
7039:parasite
6586:′
6551:tent map
6441:′
6217:′
5653:has the
5306:rational
5109:, while
2613:sequence
2559:and not
2249:operator
1171:varies.
849:Examples
441:sequence
64:sequence
40:equation
8642:Algebra
8596:, 2001
8349:2027658
8039:, real
7286:ecology
6519:chaotic
3550:Solving
3289:inverse
529:where
371:special
185:of the
8567:
8475:
8471:–568.
8447:
8407:
8391:
8377:, and
8347:
8300:
8264:
8159:
7738:, and
7644:where
7527:(IIR)
6852:
6805:
6802:
6758:
6755:
6635:
6632:
6549:, and
6487:where
5784:stable
4197:Then
3264:first
2611:Given
2286:, as
2247:is an
1555:; the
790:where
421:tuples
177:, the
38:is an
8565:S2CID
8547:arXiv
8503:(PDF)
8492:(PDF)
8345:JSTOR
8183:(PDF)
8176:(PDF)
8016:etc.
7204:with
6685:with
6517:In a
6381:with
6195:unity
5792:unity
4105:Let
1501:etc.
1258:with
655:order
148:order
8626:OEIS
8473:ISBN
8445:ISBN
8405:ISBN
8389:ISBN
8298:ISBN
8262:ISBN
8204:link
8157:ISBN
8028:and
7558:is:
7459:the
7024:The
6466:<
6298:>
6287:for
6242:<
5288:the
5105:the
3472:and
2719:is
2711:The
2671:the
2476:and
2239:The
916:>
858:The
508:>
34:, a
8557:doi
8469:551
8041:GDP
7519:In
7478:log
5308:or
2683:is
2679:of
908:for
761:for
500:for
423:of
381:).
325:of
173:In
30:In
8638::
8608:.
8592:,
8586:,
8563:.
8555:.
8543:10
8541:.
8494:.
8381:.
8373:,
8369:,
8284:;
8250:;
8246:;
8200:}}
8196:{{
8178:.
7691:,
7531:.
7511:.
7358:.
7278:.
6553:.
6545:,
6336::=
6245:1.
5941:,
5767:0.
5524:,
5491:.
5304:,
3546:.
3236:A
2373:.
1319:1.
980:)
957:1.
643:.
435:A
427:.
408:.
209:,
8623:.
8614:.
8571:.
8559::
8549::
8531:.
8521:.
8512:.
8481:.
8453:.
8413:.
8351:.
8306:.
8270:.
8206:)
8192:.
7999:T
7996:2
7990:t
7986:y
7980:2
7972:+
7967:T
7961:t
7957:x
7953:)
7948:2
7934:(
7931:+
7926:t
7922:x
7918:)
7909:1
7906:(
7903:=
7898:t
7894:y
7872:)
7867:T
7864:2
7858:t
7854:y
7847:+
7842:T
7836:t
7832:x
7828:)
7819:1
7816:(
7813:(
7807:+
7802:t
7798:x
7794:)
7785:1
7782:(
7779:=
7774:t
7770:y
7726:t
7704:t
7700:y
7679:t
7657:t
7653:x
7629:,
7624:T
7618:t
7614:y
7607:+
7602:t
7598:x
7594:)
7585:1
7582:(
7579:=
7574:t
7570:y
7546:T
7499:)
7496:)
7493:n
7490:(
7482:2
7474:(
7471:O
7442:2
7438:/
7434:n
7430:c
7426:+
7423:1
7420:=
7415:n
7411:c
7389:1
7386:=
7381:1
7377:c
7346:n
7323:n
7266:t
7244:t
7240:P
7217:t
7213:N
7189:,
7186:)
7179:t
7175:P
7171:a
7164:e
7157:1
7154:(
7149:t
7145:N
7141:=
7136:1
7133:+
7130:t
7126:P
7100:t
7096:P
7092:a
7085:e
7079:t
7075:N
7068:=
7063:1
7060:+
7057:t
7053:N
6981:h
6978:n
6975:+
6970:0
6966:t
6962:=
6957:n
6953:t
6949:,
6946:)
6941:n
6937:y
6933:,
6928:n
6924:t
6920:(
6917:f
6914:h
6911:+
6906:n
6902:y
6898:=
6893:1
6890:+
6887:n
6883:y
6849:,
6846:)
6843:h
6840:2
6837:+
6832:0
6828:t
6824:(
6821:y
6818:=
6813:2
6809:y
6799:,
6796:)
6793:h
6790:+
6785:0
6781:t
6777:(
6774:y
6771:=
6766:1
6762:y
6752:,
6749:)
6744:0
6740:t
6736:(
6733:y
6730:=
6725:0
6721:y
6697:h
6670:,
6665:0
6661:y
6657:=
6654:)
6649:0
6645:t
6641:(
6638:y
6629:,
6626:)
6623:)
6620:t
6617:(
6614:y
6611:,
6608:t
6605:(
6602:f
6599:=
6596:)
6593:t
6590:(
6583:y
6529:x
6496:x
6472:,
6469:1
6462:|
6458:)
6449:x
6445:(
6438:g
6433:|
6409:k
6389:f
6366:)
6363:x
6360:(
6357:f
6345:f
6339:f
6333:)
6330:x
6327:(
6324:g
6301:1
6295:k
6275:k
6262:f
6238:|
6234:)
6225:x
6221:(
6214:f
6209:|
6175:x
6154:f
6128:x
6101:x
6069:.
6066:)
6061:1
6055:n
6051:x
6047:(
6044:f
6041:=
6036:n
6032:x
5996:A
5970:x
5949:x
5929:A
5909:x
5886:]
5877:x
5868:1
5862:t
5858:x
5854:[
5851:A
5848:=
5845:]
5836:x
5827:t
5823:x
5819:[
5764:=
5759:0
5749:d
5745:c
5730:2
5724:d
5714:2
5710:c
5701:1
5695:d
5685:1
5681:c
5672:d
5638:,
5633:d
5627:n
5623:a
5617:d
5613:c
5609:+
5603:+
5598:2
5592:n
5588:a
5582:2
5578:c
5574:+
5569:1
5563:n
5559:a
5553:1
5549:c
5545:=
5540:n
5536:a
5512:d
5477:t
5473:x
5450:t
5446:x
5423:t
5419:w
5394:d
5391:+
5386:t
5382:w
5378:c
5373:b
5370:+
5365:t
5361:w
5357:a
5350:=
5345:1
5342:+
5339:t
5335:w
5273:)
5270:z
5267:;
5264:b
5261:,
5258:n
5255:(
5252:M
5249:=
5244:n
5240:M
5213:0
5210:=
5205:1
5202:+
5199:n
5195:M
5191:n
5183:n
5179:M
5175:)
5172:z
5169:+
5166:b
5160:n
5157:2
5154:(
5151:+
5146:1
5140:n
5136:M
5132:)
5129:n
5123:b
5120:(
5090:,
5087:)
5084:z
5081:(
5076:n
5072:J
5068:=
5063:n
5059:J
5030:1
5024:n
5020:J
5011:n
5007:J
5001:z
4997:n
4994:2
4988:=
4983:1
4980:+
4977:n
4973:J
4925:0
4919:h
4897:h
4894:n
4890:g
4886:h
4883:+
4878:n
4874:a
4870:)
4865:h
4862:n
4858:f
4854:h
4851:+
4848:1
4845:(
4842:=
4837:1
4834:+
4831:n
4827:a
4802:)
4793:k
4789:f
4783:m
4778:0
4775:=
4772:k
4761:m
4757:g
4749:1
4743:n
4738:0
4735:=
4732:m
4724:+
4719:0
4715:A
4710:(
4705:)
4699:k
4695:f
4689:1
4683:n
4678:0
4675:=
4672:k
4663:(
4659:=
4654:n
4650:a
4623:k
4619:f
4613:m
4608:0
4605:=
4602:k
4591:m
4587:g
4579:1
4573:n
4568:0
4565:=
4562:m
4554:+
4549:0
4545:A
4541:=
4533:k
4529:f
4523:1
4517:n
4512:0
4509:=
4506:k
4495:n
4491:a
4462:k
4458:f
4452:m
4447:0
4444:=
4441:k
4430:m
4426:g
4418:1
4412:n
4407:0
4404:=
4401:m
4393:=
4388:0
4384:A
4375:n
4371:A
4367:=
4364:)
4359:m
4355:A
4346:1
4343:+
4340:m
4336:A
4332:(
4327:1
4321:n
4316:0
4313:=
4310:m
4279:k
4275:f
4269:n
4264:0
4261:=
4258:k
4247:n
4243:g
4237:=
4232:n
4228:A
4219:1
4216:+
4213:n
4209:A
4182:,
4174:k
4170:f
4164:1
4158:n
4153:0
4150:=
4147:k
4136:n
4132:a
4126:=
4121:n
4117:A
4085:k
4081:f
4075:n
4070:0
4067:=
4064:k
4053:n
4049:g
4043:=
4035:k
4031:f
4025:1
4019:n
4014:0
4011:=
4008:k
3997:n
3993:a
3979:k
3975:f
3969:n
3964:0
3961:=
3958:k
3947:1
3944:+
3941:n
3937:a
3906:k
3902:f
3896:n
3891:0
3888:=
3885:k
3874:n
3870:g
3864:=
3856:k
3852:f
3846:n
3841:0
3838:=
3835:k
3823:n
3819:a
3813:n
3809:f
3794:k
3790:f
3784:n
3779:0
3776:=
3773:k
3762:1
3759:+
3756:n
3752:a
3726:n
3722:g
3718:=
3713:n
3709:a
3703:n
3699:f
3690:1
3687:+
3684:n
3680:a
3653:,
3650:0
3642:n
3638:f
3633:,
3628:n
3624:g
3620:+
3615:n
3611:a
3605:n
3601:f
3597:=
3592:1
3589:+
3586:n
3582:a
3530:n
3510:n
3449:,
3444:n
3440:a
3436:8
3428:1
3425:+
3422:n
3418:a
3414:4
3411:=
3406:2
3403:+
3400:n
3396:a
3392:3
3366:0
3363:=
3358:n
3354:a
3350:7
3347:+
3342:n
3338:a
3331:2
3328:+
3323:n
3319:a
3313:2
3305:3
3285:k
3281:k
3277:k
3273:k
3262:k
3258:k
3250:k
3246:k
3221:.
3218:)
3213:n
3209:a
3205:(
3200:k
3189:)
3184:k
3181:k
3176:(
3170:+
3164:+
3159:n
3155:a
3145:)
3140:1
3137:k
3132:(
3126:+
3121:n
3117:a
3113:=
3108:k
3105:+
3102:n
3098:a
3071:.
3066:t
3060:k
3057:+
3054:n
3050:a
3043:)
3038:t
3035:k
3030:(
3022:t
3018:)
3014:1
3008:(
3003:k
2998:0
2995:=
2992:t
2984:=
2979:n
2975:a
2969:k
2941:,
2936:1
2930:k
2916:=
2911:k
2893:k
2876:.
2871:n
2867:a
2863:+
2858:1
2855:+
2852:n
2848:a
2844:2
2836:2
2833:+
2830:n
2826:a
2822:=
2817:n
2813:a
2807:2
2779:.
2776:)
2773:a
2767:(
2761:=
2758:a
2755:)
2743:(
2740:=
2737:a
2732:2
2697:.
2694:a
2681:a
2659:,
2653:N
2646:n
2642:)
2636:n
2632:a
2628:(
2625:=
2622:a
2597:.
2592:n
2588:a
2547:,
2544:a
2531:n
2515:n
2511:a
2487:a
2464:f
2438:.
2433:n
2429:a
2420:1
2417:+
2414:n
2410:a
2406:=
2401:n
2397:)
2393:a
2387:(
2354:.
2351:)
2348:x
2345:(
2342:f
2336:)
2333:1
2330:+
2327:x
2324:(
2321:f
2318:=
2315:)
2312:x
2309:(
2306:)
2303:f
2297:(
2270:,
2219:,
2213:)
2207:k
2201:n
2197:n
2192:(
2186:=
2180:)
2175:k
2172:n
2167:(
2144:1
2141:=
2135:)
2130:0
2127:n
2122:(
2096:,
2093:k
2089:/
2085:)
2082:1
2079:+
2076:k
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2067:(
2061:)
2055:1
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2040:(
2034:=
2028:)
2023:k
2020:n
2015:(
1986:.
1980:!
1977:)
1974:k
1968:n
1965:(
1962:!
1959:k
1954:!
1951:n
1945:=
1939:)
1934:k
1931:n
1926:(
1892:1
1889:=
1882:)
1877:n
1874:n
1869:(
1862:=
1855:)
1850:0
1847:n
1842:(
1815:,
1809:)
1804:k
1800:1
1794:n
1788:(
1782:+
1776:)
1770:1
1764:k
1759:1
1753:n
1747:(
1741:=
1735:)
1730:k
1727:n
1722:(
1696:n
1676:k
1652:)
1647:k
1644:n
1639:(
1604:.
1596:2
1592:t
1585:t
1579:1
1575:t
1543:1
1540:+
1537:t
1534:=
1529:2
1525:t
1484:2
1480:F
1476:+
1471:3
1467:F
1463:=
1458:4
1454:F
1430:1
1426:F
1422:+
1417:2
1413:F
1409:=
1404:3
1400:F
1376:0
1372:F
1368:+
1363:1
1359:F
1355:=
1350:2
1346:F
1316:=
1311:1
1307:F
1285:0
1282:=
1277:0
1273:F
1241:2
1235:n
1231:F
1227:+
1222:1
1216:n
1212:F
1208:=
1203:n
1199:F
1157:0
1153:x
1132:,
1129:r
1109:.
1106:r
1083:,
1080:)
1075:n
1071:x
1064:1
1061:(
1056:n
1052:x
1048:r
1045:=
1040:1
1037:+
1034:n
1030:x
991:n
978:n
954:=
951:!
948:0
922:,
919:0
913:n
903:!
900:)
897:1
891:n
888:(
882:n
879:=
876:!
873:n
843:k
839:k
825:X
817:k
813:X
805:N
801::
775:,
772:k
766:n
756:)
751:k
745:n
741:u
737:,
731:,
726:2
720:n
716:u
712:,
707:1
701:n
697:u
693:,
690:n
687:(
681:=
676:n
672:u
658:k
625:0
621:u
600:X
592:0
588:u
577:X
560:X
554:X
547:N
543::
514:,
511:0
505:n
495:)
490:1
484:n
480:u
476:,
473:n
470:(
464:=
459:n
455:u
396:n
357:n
333:n
309:.
306:n
282:k
260:2
254:n
250:F
246:+
241:1
235:n
231:F
227:=
222:n
218:F
193:k
179:n
158:k
134:k
114:n
94:k
74:k
50:n
20:)
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