A107443 Expansion of g.f.: (1+3*x^2)/((1-x)*(1+x+2*x^2)*(1-x+2*x^2)).

1, 1, 1, 1, -3, -3, 9, 9, -11, -11, 1, 1, 45, 45, -135, -135, 229, 229, -143, -143, -483, -483, 2025, 2025, -4139, -4139, 4321, 4321, 3597, 3597, -28071, -28071, 69829, 69829, -97199, -97199, 12285, 12285, 351945, 351945, -1104971, -1104971, 1907137, 1907137, -1301523, -1301523, -3723975, -3723975

Offset: 0

Author

Creighton Dement, May 26 2005

Keywords

• easy
• sign

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,-3,3,-4,4).

Crossrefs

Cf. A001607, A174565.

Programs

Magma

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1+3*x^2)/((1-x)*(1+3*x^2+4*x^4)) )); // G. C. Greubel, Mar 24 2024

Maple

with(gfun): seriestolist(series((3*x^2+1)/((1-x)*(2*x^2+x+1)*(2*x^2-x+1)), x=0,50));

Mathematica

CoefficientList[Series[(1+3*x^2)/((1-x)*(1+3*x^2+4*x^4)), {x,0,50}], x] (* G. C. Greubel, Mar 24 2024 *)

SageMath

def A107443_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+3*x^2)/((1-x)*(1+3*x^2+4*x^4)) ).list()
A107443_list(50) # G. C. Greubel, Mar 24 2024

Formula

a(2n) = a(2n+1) = A174565(n).

a(n) = (1 - 2*(-1)^n*A001607(n) + A001607(n+1))/2. - G. C. Greubel, Mar 24 2024