A077950 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

0,5

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).

Formula

a(n) = a(n-1) - 2*a(n-3), where a(0)=1, a(1)=1, a(2)=1. - Sergei N. Gladkovskii, Aug 21 2012

a(n) = (-1)^n * A077973(n). - 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) = Sum_{k=1..floor((n+2)/2)} binomial(n+2-2*k, k-1)*(-2)^(k-1). - Taras Goy, Sep 18 2019

Mathematica

CoefficientList[Series[1/(1-x+2x^3), {x, 0, 50}], x] (* or *) LinearRecurrence[ {1, 0, -2}, {1, 1, 1}, 50] (* Harvey P. Dale, Oct 18 2013 *)

Prog

(PARI) Vec(1/(1-x+2*x^3)+O(x^50)) \\ Charles R Greathouse IV, Sep 26 2012
(Magma) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/(1-x+2*x^3) )); // G. C. Greubel, Jul 03 2019
(Sage) (1/(1-x+2*x^3)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Jul 03 2019
(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, Jul 03 2019

Crossrefs

Cf. A077973.

Sequence in context: A327555 A346915 A077934 * A077973 A210606 A254327

Adjacent sequences: A077947 A077948 A077949 * A077951 A077952 A077953

Keyword

sign,easy

Author

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

Status

approved