A077973 Expansion of 1/(1+x-2*x^3).

1, -1, 1, 1, -3, 5, -3, -3, 13, -19, 13, 13, -51, 77, -51, -51, 205, -307, 205, 205, -819, 1229, -819, -819, 3277, -4915, 3277, 3277, -13107, 19661, -13107, -13107, 52429, -78643, 52429, 52429, -209715, 314573, -209715, -209715, 838861, -1258291, 838861, 838861, -3355443, 5033165, -3355443

Offset: 0

Author

N. J. A. Sloane, Nov 17 2002, Jun 17 2007

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 (-1,0,2).

Crossrefs

Cf. A077950, A134142.

Programs

GAP

a:=[1,-1,1];; for n in [4..50] do a[n]:=-a[n-1]+2*a[n-3]; od; a; # G. C. Greubel, Jun 24 2019

Magma

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/(1+x-2*x^3) )); // G. C. Greubel, Jun 24 2019

Mathematica

LinearRecurrence[{-1,0,2}, {1,-1,1}, 50] (* or *) CoefficientList[Series[ 1/(1+x-2x^3), {x,0,50}], x] (* Harvey P. Dale, Jun 09 2017 *)

PARI

Vec(1/(1+x-2*x^3)+O(x^50)) \\ Charles R Greathouse IV, Sep 23 2012

Sage

(1/(1+x-2*x^3)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Jun 24 2019

Formula

a(n) = -a(n-1) +2*a(n-3). - Paul Curtz, Apr 23 2008

a(n) = (-1)^n * A077950(n) = (1/5) * (2*A134142(n) + 1). - Ralf Stephan, Aug 18 2013

G.f.: G(0)/(2*(1-x^2)), where G(k)= 1 + 1/(1 - x*(k+1)/(x*(k+2) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 25 2013

a(n) = (-1)^n * Sum_{k=1..floor((n+2)/2)} binomial(n+2-2*k, k-1)*(-2)^(k-1). - Taras Goy, Sep 18 2019