A199804 G.f.: 1/(1+x+x^3).

1, -1, 1, -2, 3, -4, 6, -9, 13, -19, 28, -41, 60, -88, 129, -189, 277, -406, 595, -872, 1278, -1873, 2745, -4023, 5896, -8641, 12664, -18560, 27201, -39865, 58425, -85626, 125491, -183916, 269542, -395033, 578949, -848491, 1243524, -1822473, 2670964, -3914488, 5736961, -8407925, 12322413, -18059374, 26467299, -38789712, 56849086, -83316385

Offset: 0

Author

N. J. A. Sloane, Nov 10 2011

Keywords

• sign

Comments

There are several similar sequences already in the OEIS, but this one warrants its own entry because it is one of Hirschhorn's family.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michael D. Hirschhorn, Non-trivial intertwined second-order recurrence relations, Fibonacci Quart. 43 (2005), no. 4, 316-325. See K_n.
• Index entries for linear recurrences with constant coefficients, signature (-1, 0, -1).

Programs

Mathematica

CoefficientList[Series[1/(1+x+x^3),{x,0,50}],x] (* or *) LinearRecurrence[ {-1,0,-1},{1,-1,1},50] (* Harvey P. Dale, Feb 18 2016 *)

PARI

x='x+O('x^50); Vec(1/(1+x+x^3)) \\ G. C. Greubel, Apr 29 2017

Formula

a(n) = (-1)^n*A000930(n). - R. J. Mathar, Jul 10 2012

G.f.: 1 - x/(G(0) + x) where G(k) = 1 - x^2/(1 - x^2/(x^2 - 1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 16 2012

G.f.: Q(0)/2 , where Q(k) = 1 + 1/(1 - x*(4*k+1 + x^2)/( x*(4*k+3 + x^2) - 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 08 2013

a(0)=1, a(1)=-1, a(2)=1, a(n)=a(n-1)-a(n-3). - Harvey P. Dale, Feb 18 2016