A087453 a(n) = S(n,5), where S(n,m) = Sum_{k=0..n} binomial(n,k)*Fibonacci(m*k).
0, 5, 65, 790, 9555, 115525, 1396720, 16886585, 204161685, 2468349470, 29842764575, 360804095305, 4362182828640, 52739531723965, 637629901296505, 7709053867890950, 93203771368320795, 1126849435241369885, 13623801173086279760, 164714071462466568145
Offset: 0
Links
- Colin Barker, Table of n, a(n) for n = 0..923
- Index entries for linear recurrences with constant coefficients, signature (13,-11).
Programs
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Magma
I:=[0,5]; [n le 2 select I[n] else 13*Self(n-1)-11*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Apr 27 2015
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Mathematica
Table[Sum[Binomial[n,k]Fibonacci[5k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Sep 03 2014 *) LinearRecurrence[{13, -11}, {0, 5}, 20] (* Vincenzo Librandi, Apr 27 2015 *) -
PARI
concat(0, Vec(5*x/(11*x^2-13*x+1) + O(x^100))) \\ Colin Barker, Apr 27 2015
Formula
a(n) = 13*a(n-1)-11*a(n-2).
a(n) = (1/sqrt(5))*(((13+5*sqrt(5))/2)^n-((13-5*sqrt(5))/2)^n).
G.f.: 5*x / (11*x^2-13*x+1). - Colin Barker, Apr 27 2015