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%I #17 Sep 08 2022 08:46:24
%S 1,1,1,6,12,36,91,241,632,1651,4333,11328,29684,77678,203415,532483,
%T 1394144,3649813,9555465,25016378,65493916,171465080,448901667,
%U 1175239525,3076817368,8055212055,21088819397,55211245460,144544917748
%N Expansion of (1-x-x^2)^2*(1+x-x^2)^2/((1-3*x+x^2)*(1-x)^2*(1+x)^4).
%H Vincenzo Librandi, <a href="/A324483/b324483.txt">Table of n, a(n) for n = 0..1000</a>
%H M. Baake, J. Hermisson, P. Pleasants, <a href="http://dx.doi.org/10.1088/0305-4470/30/9/016">The torus parametrization of quasiperiodic LI-classes</a>, J. Phys. A 30 (1997), no. 9, 3029-3056. See (31).
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,6,-1,-10,-1,6,1,-1).
%F G.f.: (1-x-x^2)^2*(1+x-x^2)^2/((1-3*x+x^2)*(1-x)^2*(1+x)^4).
%F From _Colin Barker_, Mar 13 2019: (Start)
%F a(n) = 2^(-4-n)*(-768*sqrt(5)*((3-sqrt(5))^n - (3+sqrt(5))^n) - 5*(517*(-2)^n + 75*2^n)*n + 25*(-1)^n*2^(1+n)*n^3) / 375 for n>1.
%F a(n) = a(n-1) + 6*a(n-2) - a(n-3) - 10*a(n-4) - a(n-5) + 6*a(n-6) + a(n-7) - a(n-8) for n>8.
%F (End)
%t CoefficientList[Series[(1 - x - x^2)^2 (1 + x - x^2)^2 / ((1 - 3 x + x^2) (1 - x)^2 (1 + x)^4), {x, 0, 33}], x] (* _Vincenzo Librandi_, Mar 13 2019 *)
%t LinearRecurrence[{1,6,-1,-10,-1,6,1,-1},{1,1,1,6,12,36,91,241,632},30] (* _Harvey P. Dale_, Sep 06 2019 *)
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1-x-x^2)^2*(1+x-x^2)^2/((1-3*x+x^2)*(1-x)^2*(1+x)^4)); // _Vincenzo Librandi_, Mar 13 2019
%o (PARI) Vec((1 + x - x^2)^2*(1 - x - x^2)^2 / ((1 - x)^2*(1 + x)^4*(1 - 3*x + x^2)) + O(x^30)) \\ _Colin Barker_, Mar 13 2019
%K nonn,easy
%O 0,4
%A _N. J. A. Sloane_, Mar 12 2019