A324483 Expansion of (1-x-x^2)^2*(1+x-x^2)^2/((1-3*x+x^2)*(1-x)^2*(1+x)^4).

1, 1, 1, 6, 12, 36, 91, 241, 632, 1651, 4333, 11328, 29684, 77678, 203415, 532483, 1394144, 3649813, 9555465, 25016378, 65493916, 171465080, 448901667, 1175239525, 3076817368, 8055212055, 21088819397, 55211245460, 144544917748

Offset: 0

Author

N. J. A. Sloane, Mar 12 2019

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• M. Baake, J. Hermisson, P. Pleasants, The torus parametrization of quasiperiodic LI-classes, J. Phys. A 30 (1997), no. 9, 3029-3056. See (31).
• Index entries for linear recurrences with constant coefficients, signature (1,6,-1,-10,-1,6,1,-1).

Programs

Magma

m:=30; R:=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

Mathematica

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 *)
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 *)

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

Formula

G.f.: (1-x-x^2)^2*(1+x-x^2)^2/((1-3*x+x^2)*(1-x)^2*(1+x)^4).

From Colin Barker, Mar 13 2019: (Start)

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.

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.

(End)