A077972 Expansion of 1/(1+x-x^2+2*x^3).

1, -1, 2, -5, 9, -18, 37, -73, 146, -293, 585, -1170, 2341, -4681, 9362, -18725, 37449, -74898, 149797, -299593, 599186, -1198373, 2396745, -4793490, 9586981, -19173961, 38347922, -76695845, 153391689, -306783378, 613566757, -1227133513, 2454267026, -4908534053, 9817068105

Offset: 0

Author

N. J. A. Sloane, Nov 17 2002

Keywords

• sign
• easy

Links

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

Crossrefs

Cf. A077947.

Programs

GAP

a:=[1,-1,2];; for n in [4..40] do a[n]:=-a[n-1]+a[n-2]-2*a[n-3]; od; a; # G. C. Greubel, Jun 24 2019

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1+x-x^2+2*x^3) )); // G. C. Greubel, Jun 24 2019

Mathematica

LinearRecurrence[{-1,1,-2}, {1,-1,2}, 40] (* or *) CoefficientList[ Series[1/(1+x-x^2+2*x^3), {x,0,40}], x] (* G. C. Greubel, Jun 24 2019 *)

PARI

Vec(1/(1+x-x^2+2*x^3)+O(x^40)) \\ Charles R Greathouse IV, Sep 27 2012

Python

def A077972(n): return -(((4<Chai Wah Wu, Apr 22 2025

Sage

(1/(1+x-x^2+2*x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jun 24 2019

Formula

a(n) = (-1)^n * A077947(n).

G.f.: Q(0)/2 , where Q(k) = 1 + 1/(1 - x*(4*k+1 - x + 2*x^2 )/( x*(4*k+3 - x + 2*x^2 ) - 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 09 2013