A096980 - OEIS

A096980

Expansion of (1+3*x)/(1-2*x-7*x^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, 118641032668801, 454208549860805, 1738904328403217

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OFFSET

0,2

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

Table of n, a(n) for n=0..26.

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

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

E.g.f.: exp(x) * (cosh(2*sqrt(2)*x) + sqrt(2)*sinh(2*sqrt(2)*x)). - Amiram Eldar, Jan 25 2026

MATHEMATICA

LinearRecurrence[{2, 7}, {1, 5}, 27] (* Amiram Eldar, Jan 25 2026 *)

PROG

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

CROSSREFS

Cf. A002315, A015519, A094015, A108051.

Sequence in context: A162270 A146790 A149696 * A339684 A149697 A149698

Adjacent sequences: A096977 A096978 A096979 * A096981 A096982 A096983

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Jul 17 2004

EXTENSIONS

More terms from Amiram Eldar, Jan 25 2026

STATUS

approved