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
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
KEYWORD
nonn,easy
AUTHOR
Ilya Gutkovskiy, Mar 03 2016
STATUS
approved