A132357 a(n) = 3*a(n-1) - a(n-3) + 3*a(n-4), with initial values 1,4,14,41.
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
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).
Programs
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Mathematica
LinearRecurrence[{3,0,-1,3},{1,4,14,41},50] (* Paolo Xausa, Dec 05 2023 *) -
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
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