A111262 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.

3, 12, 65, 403, 2652, 17889, 121859, 833260, 5706081, 39096531, 267936188, 1836369217, 12586419075, 86267964108, 591287758337, 4052742230419, 27777897084444, 190392509164065, 1304969593244291, 8944394450283436

Offset: 1

Author

Benoit Cloitre, Nov 12 2005, corrected Feb 24 2008

Keywords

• nonn

Comments

Values are always integers.

Links

• Indranil Ghosh, Table of n, a(n) for n = 1..1194
• Index entries for linear recurrences with constant coefficients, signature (10, -23, 10, -1).

Crossrefs

Cf. A001519.

Programs

Mathematica

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 *)

PARI

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

Formula

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

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

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

a(n) = (Lucas(2*n-1)+2)*Fibonacci(2*n-1) = A162483(n-1)*A001519(n). - Ehren Metcalfe, Jun 04 2019