A135266 Partial sums of A132357.

0, 1, 5, 19, 60, 182, 546, 1639, 4919, 14761, 44286, 132860, 398580, 1195741, 3587225, 10761679, 32285040, 96855122, 290565366, 871696099, 2615088299, 7845264901, 23535794706, 70607384120, 211822152360, 635466457081

Offset: 0

Author

Paul Curtz, Dec 02 2007

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-3,-1,4,-3).

Programs

Mathematica

Join[{0}, Table[(1/4)*3^(n + 1) - (1/12)*(-1)^n + (1/3)*Cos[Pi*n/3] - (Sqrt[3]/3)*Sin[Pi*n/3] - 1, {n, 1, 25}]] (* G. C. Greubel, Oct 07 2016 *)

PARI

a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; -3,4,-1,-3,4]^n*[0;1;5;19;60])[1,1] \\ Charles R Greathouse IV, Oct 08 2016

Formula

a(n+1) - 3*a(n) = 0, 1, 2, 4, 3, 2,... (periodically extended with period length 6) = partial sums of A132367.

a(n) = (1/4)*3^(n+1) - (1/12)*(-1)^n + (1/3)*cos(Pi*n/3) - (sqrt(3)/3)*sin (Pi*n/3) - 1. Or, a(n) = (1/4)*3^(n+1) + (1/4)*[ -3; -5; -7; -5; -3; -1] for n>=0. - Richard Choulet, Jan 02 2008

O.g.f.: x*(1 +x +2*x^2)/((3*x-1)*(x+1)(x^2-x+1)*(x-1)). - R. J. Mathar, Jul 28 2008

Extensions

Edited and extended by R. J. Mathar, Jul 28 2008