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

%I A269500 #48 Feb 16 2025 08:33:30
%S A269500 0,55,6765,832040,102334155,12586269025,1548008755920,190392490709135,
%T A269500 23416728348467685,2880067194370816120,354224848179261915075,
%U A269500 43566776258854844738105,5358359254990966640871840,659034621587630041982498215,81055900096023504197206408605
%N A269500 a(n) = Fibonacci(10*n).
%C A269500 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 A269500 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>
%H A269500 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (123,-1)
%F A269500 G.f.: 55*x/(1 - 123*x + x^2).
%F A269500 a(n) = 123*a(n-1) - a(n-2).
%F A269500 a(n) = A000045(10*n).
%F A269500 Lim_{n -> infinity} a(n + 1)/a(n) = phi^10 = 122.9918693812442…
%t A269500 Fibonacci[10Range[0, 14]]
%t A269500 FullSimplify[Table[(((1 + Sqrt[5])/2)^(10 n) - (2/(1 + Sqrt[5]))^(10 n))/Sqrt[5], {n, 0, 12}]]
%t A269500 LinearRecurrence[{123, -1}, {0, 55}, 15]
%o A269500 (PARI) a(n) = fibonacci(10*n); \\ _Michel Marcus_, Mar 03 2016
%o A269500 (PARI) concat(0, Vec(55*x/(1-123*x+x^2) + O(x^100))) \\ _Altug Alkan_, Mar 04 2016
%Y A269500 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 A269500 Cf. A000032 (Lucas numbers), A001622 (golden ratio).
%K A269500 nonn,easy
%O A269500 0,2
%A A269500 _Ilya Gutkovskiy_, Mar 03 2016