A087426 a(n) = S(n,4) where S(n,m) = Sum_{k=0..n} binomial(n,k)*Fibonacci(m*k).

0, 3, 27, 216, 1701, 13365, 104976, 824499, 6475707, 50860872, 399466485, 3137450517, 24641856288, 193539651939, 1520080160859, 11938864580280, 93769059774789, 736471756750581, 5784324272782128, 45430672644283923

Offset

0,2

Links

Table of n, a(n) for n=0..19.

M. Griffiths, Families of Sequences From a Class of Multinomial Sums, J. Int. Seq. 15 (2012) # 12.1.8.

Index entries for linear recurrences with constant coefficients, signature (9,-9).

Formula

a(n) = 9*a(n-1)-9*a(n-2).

a(n) = (1/sqrt(5))*(((9+3*sqrt(5))/2)^n-((9-3*sqrt(5))/2)^n).

a(n) = 3^n*F(2n). - Benoit Cloitre, Sep 13 2005

G.f.: 3*x / (9*x^2-9*x+1). - Colin Barker, Jun 26 2013

E.g.f.: 2*exp(9*x/2)*sinh(3*sqrt(5)*x/2)/sqrt(5). - Stefano Spezia, Feb 23 2025

Sum_{k=0..n+1} a(k) = 3+27*A057085(n). - Greg Dresden and Xiaoya Gao, Nov 25 2025

Crossrefs

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

Sequence in context: A198686 A357662 A377242 * A145608 A083713 A230179

Adjacent sequences: A087423 A087424 A087425 * A087427 A087428 A087429

Keyword

nonn,easy

Author

Benoit Cloitre, Oct 23 2003

Status

approved