A077898 Expansion of (1 - x)^(-1)/(1 + x - 2*x^2).

1, 0, 3, -2, 9, -12, 31, -54, 117, -224, 459, -906, 1825, -3636, 7287, -14558, 29133, -58248, 116515, -233010, 466041, -932060, 1864143, -3728262, 7456549, -14913072, 29826171, -59652314, 119304657, -238609284, 477218599, -954437166, 1908874365, -3817748696, 7635497427

Offset

0,3

Comments

Partial sums of A077925 (signed Jacobsthal numbers). - Paul Barry, Aug 26 2003

The generalized (3,-2)-Padovan sequence p(3,-2;n). See the W. Lang link under A000931 with (A,B)=(3,-2). - Wolfdieter Lang, Jun 28 2010

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (0,3,-2).

Formula

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

a(n) = Sum_{k=0..n} Sum_{j=0..k} Sum_{i=0..j} binomial(j, i)*(-3)^i. - Paul Barry, Aug 26 2003

a(n) = (-1)^n * A053088(n). - R. J. Mathar, Aug 30 2008

From Colin Barker, Apr 21 2016: (Start)

a(n) = 3*a(n-2) - 2*a(n-3) for n>2.

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

E.g.f.: (4*exp(-2*x) + (5 + 3*x)*exp(x))/9. - Ilya Gutkovskiy, Apr 21 2016

a(n) = Sum_{k=0..n} (n+1-k)*(-2)^k. - Bruno Berselli, May 15 2018

Example

(3,-2)-Padovan combinatorics from the (3,2)-Morse code with weights -2 and 3 for 3-lines -- and 2-lines -, respectively (see the W. Lang link under A000931). n=5: two codes - -- and -- - with the weights (3^1)*(-2)^1 and (-2)^1*3^1, respectively, adding up to 2*(3)(-2) = -12 = a(5). - Wolfdieter Lang, Jun 28 2010

Maple

A077898:=n->(5+(-1)^n*2^(2+n)+3*n)/9: seq(A077898(n), n=0..50); # Wesley Ivan Hurt, Apr 21 2016

Mathematica

CoefficientList[Series[(1 - x)^(-1)/(1 + x - 2*x^2), {x, 0, 40}], x] (* Wesley Ivan Hurt, Apr 21 2016 *)

Prog

(PARI) Vec(1/(1-x)/(1+x-2*x^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012
(Magma) [(5+(-1)^n*2^(2+n)+3*n)/9 : n in [0..50]]; // Wesley Ivan Hurt, Apr 21 2016

Crossrefs

Cf. A000931, A053088, A077925.

Sequence in context: A303901 A053151 A053088 * A303631 A358603 A076584

Adjacent sequences: A077895 A077896 A077897 * A077899 A077900 A077901

Keyword

sign,easy

Author

N. J. A. Sloane, Nov 17 2002

Status

approved