A078033 Expansion of (1-x) / (1+x^2+2*x^3).

1, -1, -1, -1, 3, 3, -1, -9, -5, 11, 23, -1, -45, -45, 47, 135, 43, -229, -313, 143, 771, 483, -1057, -2025, 91, 4139, 3959, -4321, -12237, -3597, 20879, 28071, -13685, -69829, -42457, 97199, 182115, -12285, -376513, -351945, 401083, 1104971, 302807, -1907137, -2512749, 1301523, 6327023

Offset: 0

Author

N. J. A. Sloane, Nov 17 2002

Keywords

• sign
• easy

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,-1,-2).

Programs

GAP

a:=[1,-1,-1];; for n in [4..40] do a[n]:=-a[n-2]-2*a[n-3]; od; a; # G. C. Greubel, Nov 22 2019

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x)/((1+x)*(1-x+2*x^2)) )); // G. C. Greubel, Nov 22 2019

Maple

seq(coeff(series((1-x)/((1+x)*(1-x+2*x^2)), x, n+1), x, n), n = 0..40); # G. C. Greubel, Nov 22 2019

Mathematica

LinearRecurrence[{0,-1,-2}, {1,-1,-1}, 40] (* G. C. Greubel, Nov 22 2019 *)
CoefficientList[Series[(1-x)/(1+x^2+2x^3),{x,0,60}],x] (* Harvey P. Dale, Mar 31 2023 *)

PARI

Vec((1-x)/((1+x)*(1-x+2*x^2)) + O(x^40)) \\ Colin Barker, May 18 2019

Sage

def A078033_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)/((1+x)*(1-x+2*x^2)) ).list()
A078033_list(40) # G. C. Greubel, Nov 22 2019

Formula

2*((-1)^n)*a(n) + A001607(n+2) = 1 - Creighton Dement, Oct 30 2004

a(n) = -a(n-2) - 2*a(n-3) for n>2. - Colin Barker, May 18 2019