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

1,1

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).

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

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 *)

Prog

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

Crossrefs

Cf. A001519.

Sequence in context: A302195 A359660 A196559 * A139134 A216373 A200309

Adjacent sequences: A111259 A111260 A111261 * A111263 A111264 A111265

Keyword

nonn

Author

Benoit Cloitre, Nov 12 2005, corrected Feb 24 2008

Status

approved