A103819 Whitney transform of Jacobsthal numbers.

0, 1, 3, 8, 23, 63, 172, 471, 1287, 3516, 9607, 26247, 71708, 195911, 535239, 1462300, 3995079, 10914759, 29819676, 81468871, 222577095, 608091932, 1661338055, 4538859975, 12400396060, 33878512071, 92557816263, 252872656668

Offset: 0

Author

Paul Barry, Feb 16 2005

Keywords

• easy
• nonn

Comments

The Whitney transform maps the sequence with g.f. g(x) to that with g.f. (1/(1-x))*g(x(1+x)).

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,2,1,-2,-2).

Crossrefs

Cf. A004070, A001045, A002605, A099837.

Programs

Mathematica

LinearRecurrence[{2,2,1,-2,-2},{0,1,3,8,23},30] (* Harvey P. Dale, Nov 02 2024 *)

Formula

G.f.: x(1+x)/((1-x)(1+x+x^2)(1-2x-2x^2)).

a(n) = 2a(n-1)+2a(n-2)+a(n-3)-2a(n-4)-2a(n-5).

a(n) = Sum_{k=0..n} Sum_{i=0..n} C(k, i-k)*A001045(k).

9*a(n) = -2 +2*(A002605(n)+2*A002605(n+1))-A099837(n+3). - R. J. Mathar, Oct 23 2011