A132357 a(n) = 3*a(n-1) - a(n-3) + 3*a(n-4).

1, 4, 14, 41, 122, 364, 1093, 3280, 9842, 29525, 88574, 265720, 797161, 2391484, 7174454, 21523361, 64570082, 193710244, 581130733, 1743392200, 5230176602, 15690529805, 47071589414, 141214768240, 423644304721

Offset

0,2

Links

Paolo Xausa, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,0,-1,3).

Formula

O.g.f.: -(1+x+2*x^2)/((3*x-1)*(x+1)*(x^2-x+1)) = -(3/2)/(3*x-1)+(1/3)*(x-2)/(x^2-x+1)+(1/ 6)/(x+1). - R. J. Mathar, Nov 28 2007

a(n) = (1/2)*3^(n+1) + (1/6)*(-1)^n - (2/3)*cos(Pi*n/3). Or, a(n) = (1/2)*3^(n+1) + (1/2)*[ -1; -1; 1; 1; 1; -1]. - Richard Choulet, Jan 02 2008

a(n+1) - 3a(n) = A132367(n+1). - Paul Curtz, Dec 02 2007

6*a(n) = (-1)^n +3^(n+2) -2*A057079(n+1). - R. J. Mathar, Oct 03 2021

Mathematica

LinearRecurrence[{3, 0, -1, 3}, {1, 4, 14, 41}, 50] (* Paolo Xausa, Dec 05 2023 *)

Prog

(PARI) a(n)=([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; 3, -1, 0, 3]^n*[1; 4; 14; 41])[1, 1] \\ Charles R Greathouse IV, Oct 08 2016

Crossrefs

First differences of A132353.

Cf. A129339.

Sequence in context: A196713 A358587 A237853 * A262875 A219867 A295201

Adjacent sequences: A132354 A132355 A132356 * A132358 A132359 A132360

Keyword

nonn,easy

Author

Paul Curtz, Nov 24 2007

Status

approved