A135365 a(n) = 3*a(n-1) - a(n-3) + 3*a(n-4), with initial values 1,3,9,11.
1, 3, 9, 11, 33, 99, 313, 939, 2817, 8435, 25305, 75915, 227761, 683283, 2049849, 6149531, 18448593, 55345779, 166037353, 498112059, 1494336177, 4483008515, 13449025545, 40347076635, 121041229921, 363123689763, 1089371069289, 3268113207851, 9804339623553
Offset: 0
Links
- Colin Barker, 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
Join[{1, 3}, LinearRecurrence[{3, 0, -1, 3}, {9, 11, 33, 99}, 25]] (* G. C. Greubel, Oct 11 2016 *) -
PARI
Vec((1-15*x^3)/((1+x)*(1-3*x)*(1-x+x^2)) + O(x^40)) \\ Colin Barker, Feb 10 2016
Formula
From Richard Choulet, Jan 02 2008: (Start)
a(n) = (1/7)*3^(n+1) + (4/3)*(-1)^n - (16/21)*cos(Pi*n/3) + (16*sqrt(3)/7)*sin(Pi*n/3).
a(n) = (1/7)*3^(n+1) + (1/7)*[4; 12; 36; -4; -12; -36] for n>=0. (End)
G.f.: (1 - 15*x^3) / ((1+x)*(1-3*x)*(1-x+x^2)). - Colin Barker, Feb 10 2016