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a(n) = S(n,5), where S(n,m) = Sum_{k=0..n} binomial(n,k)*Fibonacci(m*k).
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%I #19 Aug 24 2024 21:43:45

%S 0,5,65,790,9555,115525,1396720,16886585,204161685,2468349470,

%T 29842764575,360804095305,4362182828640,52739531723965,

%U 637629901296505,7709053867890950,93203771368320795,1126849435241369885,13623801173086279760,164714071462466568145

%N a(n) = S(n,5), where S(n,m) = Sum_{k=0..n} binomial(n,k)*Fibonacci(m*k).

%H Colin Barker, <a href="/A087453/b087453.txt">Table of n, a(n) for n = 0..923</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (13,-11).

%F a(n) = 13*a(n-1)-11*a(n-2).

%F a(n) = (1/sqrt(5))*(((13+5*sqrt(5))/2)^n-((13-5*sqrt(5))/2)^n).

%F G.f.: 5*x / (11*x^2-13*x+1). - _Colin Barker_, Apr 27 2015

%t Table[Sum[Binomial[n,k]Fibonacci[5k],{k,0,n}],{n,0,20}] (* _Harvey P. Dale_, Sep 03 2014 *)

%t LinearRecurrence[{13, -11}, {0, 5}, 20] (* _Vincenzo Librandi_, Apr 27 2015 *)

%o (PARI) concat(0, Vec(5*x/(11*x^2-13*x+1) + O(x^100))) \\ _Colin Barker_, Apr 27 2015

%o (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

%Y Cf. A001906 (S(n,1)), A030191 (S(n,2)).

%K nonn,easy

%O 0,2

%A _Benoit Cloitre_, Oct 23 2003