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%I #14 Sep 09 2016 18:05:54
%S 0,6,4,22,20,86,84,342,340,1366,1364,5462,5460,21846,21844,87382,
%T 87380,349526,349524,1398102,1398100,5592406,5592404,22369622,
%U 22369620,89478486,89478484,357913942,357913940,1431655766,1431655764,5726623062
%N Expansion of -2*x*(-3-2*x+4*x^2) / ((x-1)*(2*x+1)*(2*x-1)*(1+x)).
%C 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].
%H Colin Barker, <a href="/A120462/b120462.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,5,0,-4).
%F a(2*n+1) = A047849(n+2). a(2*n)= 2*A020988(n). - _R. J. Mathar_, Nov 07 2011
%F From _Colin Barker_, Sep 09 2016: (Start)
%F a(n) = -2*(1/6 + (-2)^n/3 + (-1)^n/2 - 2^n).
%F a(n) = 5*a(n-2)-4*a(n-4) for n>3.
%F (End)
%t 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}]
%o (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
%K nonn,easy
%O 0,2
%A _Roger L. Bagula_ and _Gary W. Adamson_, Jun 28 2006
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