A077990 Expansion of 1/(1 + 2*x + x^2 - x^3).

1, -2, 3, -3, 1, 4, -12, 21, -26, 19, 9, -63, 136, -200, 201, -66, -269, 805, -1407, 1740, -1268, -611, 4230, -9117, 13393, -13439, 4368, 18096, -53999, 94270, -116445, 84621, 41473, -284012, 611172, -896859, 898534, -289037, -1217319, 3622209, -6316136, 7792744, -5647143, -2814594

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 (-2,-1,1).

Crossrefs

Cf. A077941.

Programs

GAP

a:=[1,-2,3];; for n in [4..50] do a[n]:=-2*a[n-1]-a[n-2]+a[n-3]; od; a; # G. C. Greubel, Jun 26 2019

Magma

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/(1+2*x+x^2-x^3) )); // G. C. Greubel, Jun 26 2019

Maple

A077990 := proc(n)
    option remember ;
    if n <=2 then
        (-1)^n*(n+1) ;
    else
        -2*procname(n-1)-procname(n-2)+procname(n-3) ;
    end if;
end proc:
seq(A077990(n),n=0..20) ; # R. J. Mathar, Feb 25 2024

Mathematica

CoefficientList[Series[1/(1+2x+x^2-x^3),{x,0,50}],x] (* or *) LinearRecurrence[ {-2,-1,1},{1,-2,3},50] (* Harvey P. Dale, Aug 10 2016 *)

PARI

Vec(1/(1+2*x+x^2-x^3)+O(x^50)) \\ Charles R Greathouse IV, Sep 26 2012

Sage

(1/(1+2*x+x^2-x^3)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Jun 26 2019

Formula

From Paul Barry, May 10 2005: (Start)

G.f.: 1/((1+x)^2-x^3).

a(n) = Sum_{k=0..n+4} (-1)^(n-k-1)*C(n+3, k)*Sum_{j=0..floor(k/3)} C(k-2j, j). (End)

a(n) = (-1)^n * A077941(n). - G. C. Greubel, Jun 26 2019