%I #47 Mar 21 2016 04:01:23

%S 0,55,6765,832040,102334155,12586269025,1548008755920,190392490709135,

%T 23416728348467685,2880067194370816120,354224848179261915075,

%U 43566776258854844738105,5358359254990966640871840,659034621587630041982498215,81055900096023504197206408605

%N a(n) = Fibonacci(10*n).

%C 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).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (123,-1)

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

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

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

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

%t Fibonacci[10Range[0, 14]]

%t FullSimplify[Table[(((1 + Sqrt[5])/2)^(10 n) - (2/(1 + Sqrt[5]))^(10 n))/Sqrt[5], {n, 0, 12}]]

%t LinearRecurrence[{123, -1}, {0, 55}, 15]

%o (PARI) a(n) = fibonacci(10*n); \\ _Michel Marcus_, Mar 03 2016

%o (PARI) concat(0, Vec(55*x/(1-123*x+x^2) + O(x^100))) \\ _Altug Alkan_, Mar 04 2016

%Y 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).

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

%K nonn,easy

%O 0,2

%A _Ilya Gutkovskiy_, Mar 03 2016