A111262 - OEIS

%I #25 Mar 03 2024 14:20:18

%S 3,12,65,403,2652,17889,121859,833260,5706081,39096531,267936188,

%T 1836369217,12586419075,86267964108,591287758337,4052742230419,

%U 27777897084444,190392509164065,1304969593244291,8944394450283436

%N a(n) = (1/n)*Sum_{k=1..n} F(4*k)*B(2*n-2*k)*binomial(2*n,2*k), where F are Fibonacci numbers and B are Bernoulli numbers.

%C Values are always integers.

%H Indranil Ghosh, <a href="/A111262/b111262.txt">Table of n, a(n) for n = 1..1194</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (10, -23, 10, -1).

%F a(n) = F(4*n-2) + 2*F(2*n-1).

%F Recurrence: a(n) = 10*a(n-1) - 23*a(n-2) + 10*a(n-3) - a(n-4).

%F O.g.f.: -x*(-3+18*x-14*x^2+x^3)/((x^2-3*x+1)*(x^2-7*x+1)) = -1+(2-4*x)/(x^2-3*x+1)+(-1+8*x)/(x^2-7*x+1). - _R. J. Mathar_, Nov 23 2007

%F a(n) = (Lucas(2*n-1)+2)*Fibonacci(2*n-1) = A162483(n-1)*A001519(n). - _Ehren Metcalfe_, Jun 04 2019

%t Table[(1/n)*Sum[Fibonacci[4k]BernoulliB[2n-2k]Binomial[2n,2k],{k,1,n}],{n,1,20}] (* or *) Table[Fibonacci[4n-2]+2Fibonacci[2n-1],{n,1,20}] (* or *) LinearRecurrence[{10,-23,10,-1},{3,12,65,403},20] (* _Indranil Ghosh_, Feb 26 2017 *)

%o (PARI) a(n)=fibonacci(4*n-2)+2*fibonacci(2*n-1)

%Y Cf. A001519.

%K nonn

%O 1,1

%A _Benoit Cloitre_, Nov 12 2005, corrected Feb 24 2008