A092220 Expansion of x*(1-x)/ ((1+x)*(1-x+x^2)) in powers of x.

0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1

Offset

0,1

Comments

Multiplicative with a(2^e) = -1, a(3^e) = 0, a(p^e) = 1 otherwise. - David W. Wilson Jun 12 2005

Transform of the Jacobsthal numbers A001045 under the Riordan array A102587. - Paul Barry, Jul 14 2005

The BINOMIAL transform generates (-1)^(n+1)*A024495(n+1). - R. J. Mathar, Apr 07 2008

Links

Table of n, a(n) for n=0..94.

Michael Somos, Rational Function Multiplicative Coefficients

Index entries for linear recurrences with constant coefficients, signature (0,0,-1).

Formula

Euler transform of length 6 sequence [-1, 0, -1, 0, 0, 1]. - Michael Somos, Apr 10 2011

Moebius transform is length 6 sequence [1, -2, -1, 0, 0, 2]. - Michael Somos, Apr 10 2011

G.f.: x * (1 - x) * (1 - x^3) / (1 - x^6). - Michael Somos, Apr 10 2011

a(n) = a(-n), a(n + 3) = -a(n), a(3*n) = 0, for all n in Z. - Michael Somos, Apr 10 2011

a(n) = 2*cos(Pi*n/3)/3 - 2(-1)^n/3.

a(n) = 3*a(n-1) - a(n-3) + 3*a(n-4). - Paul Curtz, Dec 10 2007

a(n) = ( (-1)^floor((n+1)/3) - (-1)^n )/2. [Bruno Berselli, Jul 09 2013]

a(n) = S(n-1,-1), n >= 0, with Chebyshev's S-polynomials evaluated at -1 (see A049310). - Wolfdieter Lang, Sep 06 2013

a(n) = A131531(n+2) - A131531(n+1) . - R. J. Mathar, Nov 28 2019

Example

G.f. = x - x^2 - x^4 + x^5 + x^7 - x^8 - x^10 + x^11 + x^13 - x^14 - x^16 + x^17 + ...

Mathematica

a[ n_] := {1, -1, 0, -1, 1, 0}[[Mod[n, 6, 1]]]; (* Michael Somos, Aug 25 2014 *)
LinearRecurrence[{0, 0, -1}, {0, 1, -1}, 120] (* or *) PadRight[{}, 120, {0, 1, -1, 0, -1, 1}] (* Harvey P. Dale, Mar 30 2016 *)

Prog

(PARI) {a(n) = [0, 1, -1, 0, -1, 1][n%6 + 1]}; /* Michael Somos, Apr 10 2011 */

Crossrefs

Sequence in context: A094217 A280261 A174784 * A011655 A102283 A128834

Adjacent sequences: A092217 A092218 A092219 * A092221 A092222 A092223

Keyword

sign,easy,mult

Author

Paul Barry, Feb 25 2004

Status

approved