A120462 Expansion of -2*x*(-3-2*x+4*x^2) / ((x-1)*(2*x+1)*(2*x-1)*(1+x)).

0, 6, 4, 22, 20, 86, 84, 342, 340, 1366, 1364, 5462, 5460, 21846, 21844, 87382, 87380, 349526, 349524, 1398102, 1398100, 5592406, 5592404, 22369622, 22369620, 89478486, 89478484, 357913942, 357913940, 1431655766, 1431655764, 5726623062

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Jun 28 2006

Keywords

• nonn
• easy

Comments

Top element of the vector obtained by multiplying the n-th power of the 6 X 6 matrix [[0, 1, 0, 0, 0, 1], [1, 0, 1, 0, 0, 0], [0, 1, 0, 1, 0, 0], [0, 0, 1, 0, 1, 0], [0, 0, 0, 1, 0, 1], [1, 0, 0, 0, 1, 0]] by the column vector [0, 1, 1, 2, 3, 5].

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,5,0,-4).

Programs

Mathematica

M = {{0, 1, 0, 0, 0, 1}, {1, 0, 1, 0, 0, 0}, {0, 1, 0, 1, 0, 0}, {0, 0, 1, 0, 1, 0}, {0, 0, 0, 1, 0, 1}, {1, 0, 0, 0, 1, 0}} v[1] = {0, 1, 1, 2, 3, 5} v[n_] := v[n] = M.v[n - 1] a = Table[Floor[v[n][[1]]], {n, 1, 50}]
LinearRecurrence[{0,5,0,-4},{0,6,4,22},40] (* Harvey P. Dale, Jul 28 2024 *)

PARI

concat(0, Vec(2*x*(3+2*x-4*x^2)/((1-x)*(1+x)*(1-2*x)*(1+2*x)) + O(x^40))) \\ Colin Barker, Sep 09 2016

Formula

a(2*n+1) = A047849(n+2). a(2*n)= 2*A020988(n). - R. J. Mathar, Nov 07 2011

From Colin Barker, Sep 09 2016: (Start)

a(n) = -2*(1/6 + (-2)^n/3 + (-1)^n/2 - 2^n).

a(n) = 5*a(n-2)-4*a(n-4) for n>3.

(End)