A098554 G.f.: x*(1-x^2)/((1+x^2)*(1+x+x^2)).

0, 1, -1, -2, 3, 1, -4, 1, 3, -2, -1, 1, 0, 1, -1, -2, 3, 1, -4, 1, 3, -2, -1, 1, 0, 1, -1, -2, 3, 1, -4, 1, 3, -2, -1, 1, 0, 1, -1, -2, 3, 1, -4, 1, 3, -2, -1, 1, 0, 1, -1, -2, 3, 1, -4, 1, 3, -2, -1, 1, 0, 1, -1, -2, 3, 1, -4, 1, 3, -2, -1, 1, 0, 1, -1, -2, 3, 1, -4, 1, 3, -2, -1, 1, 0, 1, -1, -2, 3, 1, -4, 1, 3, -2, -1, 1, 0, 1, -1, -2, 3, 1, -4, 1, 3, -2

Offset

0,4

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

G. I. Lehrer and G. B. Segal, Homology stability for classical regular semisimple varieties, Math. Zeit., 236 (2001), 251-290; see Th. 7.12.

Index entries for linear recurrences with constant coefficients, signature (-1,-2,-1,-1).

Formula

Let b(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(0^(n-2k)-(-1)^(n-2k)). Then a(n) = b(n) - b(n-2), or a(n) = Sum_{j=0..n} b(n-j)*(binomial(1, j/2)*(-1)^(j/2)*(1+(-1)^j)/2). The g.f. is obtained from the g.f. x/(1+x) of 0^n-(-1)^n by applying the transformation G(x)->((1-x^2)/(1+x^2))G(x/(1+x^2)). - Paul Barry, Oct 26 2004

a(n) = (-1)^n*(A112553(n-1) - A112553(n-3)). - R. J. Mathar, Sep 27 2014

a(0)=0, a(1)=1, a(2)=-1, a(3)=-2, a(n) = a(n-1) - 2*a(n-2) - a(n-3) - a(n-4). - Harvey P. Dale, Jan 16 2016

Mathematica

CoefficientList[Series[x*(1-x^2)/((1+x^2)*(1+x+x^2)), {x, 0, 110}], x] (* or *) LinearRecurrence[{-1, -2, -1, -1}, {0, 1, -1, -2}, 110] (* Harvey P. Dale, Jan 16 2016 *)

Prog

(PARI) x='x+O('x^30); concat([0], Vec(x*(1-x^2)/((1+x^2)*(1+x+x^2)))) \\ G. C. Greubel, Jan 17 2018
(Magma) I:=[0, 1, -1, -2]; [n le 4 select I[n] else -Self(n-1) - 2*Self(n-2) -Self(n-3) - Self(n-4): n in [1..30]]; // G. C. Greubel, Jan 17 2018

Crossrefs

Sequence in context: A035612 A199539 A089555 * A226081 A109201 A002946

Adjacent sequences: A098551 A098552 A098553 * A098555 A098556 A098557

Keyword

sign,easy

Author

N. J. A. Sloane, Oct 26 2004

Status

approved