A269500 a(n) = Fibonacci(10*n).

0, 55, 6765, 832040, 102334155, 12586269025, 1548008755920, 190392490709135, 23416728348467685, 2880067194370816120, 354224848179261915075, 43566776258854844738105, 5358359254990966640871840, 659034621587630041982498215, 81055900096023504197206408605

Offset

0,2

Comments

More generally, the ordinary generating function for the Fibonacci(k*n) is F(k)*x/(1 - L(k)*x + (-1)^k*x^2), where F(k) is the k-th Fibonacci number (A000045), L(k) is the k-th Lucas number (A000032), or (phi^k - (-1/phi)^k)*x/(sqrt(5)*(1 - (phi^k + (-1/phi)^k)*x + (-1)^k*x^2)), where phi is the golden ratio (A001622).

Links

Table of n, a(n) for n=0..14.

Eric Weisstein's World of Mathematics, Fibonacci Number

Index entries for linear recurrences with constant coefficients, signature (123,-1)

Formula

G.f.: 55*x/(1 - 123*x + x^2).

a(n) = 123*a(n-1) - a(n-2).

a(n) = A000045(10*n).

Lim_{n -> infinity} a(n + 1)/a(n) = phi^10 = 122.9918693812442…

Mathematica

Fibonacci[10Range[0, 14]]
FullSimplify[Table[(((1 + Sqrt[5])/2)^(10 n) - (2/(1 + Sqrt[5]))^(10 n))/Sqrt[5], {n, 0, 12}]]
LinearRecurrence[{123, -1}, {0, 55}, 15]

Prog

(PARI) a(n) = fibonacci(10*n); \\ Michel Marcus, Mar 03 2016
(PARI) concat(0, Vec(55*x/(1-123*x+x^2) + O(x^100))) \\ Altug Alkan, Mar 04 2016

Crossrefs

Cf. similar sequences of the form Fibonacci(k*n): A000045 (k = 1), A001906 (k = 2), A014445 (k = 3), A033888 (k = 4), A102312 (k = 5), A134492 (k = 6), A134498 (k = 7), A138473 (k = 8), A138590 (k = 9), this sequence (k = 10), A167398 (k = 11), A214855 (k = 15).

Cf. A000032 (Lucas numbers), A001622 (golden ratio).

Sequence in context: A114049 A028471 A004708 * A090813 A145617 A105842

Adjacent sequences: A269497 A269498 A269499 * A269501 A269502 A269503

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Mar 03 2016

Status

approved