A096980 Expansion of (1+3x)/(1-2x-7x^2).

1, 5, 17, 69, 257, 997, 3793, 14565, 55681, 213317, 816401, 3126021, 11966849, 45815845, 175399633, 671510181, 2570817793, 9842206853, 37680138257, 144255724485, 552272416769, 2114334904933, 8094576727249, 30989497789029

Offset: 0

Author

Paul Barry, Jul 17 2004

Keywords

• easy
• nonn

Comments

Second binomial transform is A002315 (NSW numbers). Binomial transform of A094015.

Binomial transform is A108051 (shifted left, without leading zero). - R. J. Mathar, Jul 11 2012

Links

• Index entries for linear recurrences with constant coefficients, signature (2,7).

Crossrefs

Cf. A002315, A094015, A108051, A015519.

Programs

PARI

x='x + O('x^24); Vec((1 + 3*x)/(1 - 2*x - 7*x^2)) \\ Indranil Ghosh, Apr 11 2017

Formula

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

a(n) = 3*Sum_{k=0..floor((n-1)/2)} binomial(n-k-1, k)*(7/2)^k*2^(n-k-1) + Sum_{k=0..floor(n/2)} binomial(n-k, k)*(7/2)^k*2^(n-k).

a(n) = A015519(n+1) + 3*A015519(n). - R. J. Mathar, Jul 11 2012

Satisfies recurrence relation system a(n) = 3*a(n-1) + 2*b(n-1), b(n) = 2*a(n-1) - b(n-1), a(0)=1, b(0)=1. - Ilya Gutkovskiy, Apr 11 2017