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

Author

Paul Barry, Feb 25 2004

Keywords

• sign
• easy
• mult

Comments

Period 6: repeat [0, 1, -1, 0, -1, 1]. - Joerg Arndt, Aug 28 2024

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

Examples

			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 + ...
		

Links

• Michael Somos, Rational Function Multiplicative Coefficients, 2014.
• Index entries for linear recurrences with constant coefficients, signature (0,0,-1).

Crossrefs

Cf. A001045, A024495, A049310, A102587, A131531, A128834.

Programs

Maple

seq(2*sin(Pi*n^2/3)/sqrt(3), n=0..100); # Ridouane Oudra, Oct 30 2024

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 *)

PARI

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

Formula

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

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

From Michael Somos, Apr 10 2011: (Start)

Euler transform of length 6 sequence [-1, 0, -1, 0, 0, 1].

Moebius transform is length 6 sequence [1, -2, -1, 0, 0, 2].

G.f.: x * (1 - x) * (1 - x^3) / (1 - x^6).

a(n) = a(-n), a(n + 3) = -a(n), a(3*n) = 0, for all n in Z. (End)

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

a(n) = A128834(n^2). - Ridouane Oudra, Oct 30 2024

E.g.f.: 2*(exp(x/2)*cos(sqrt(3)*x/2) - cosh(x) + sinh(x))/3. - Stefano Spezia, Oct 31 2024

Dirichlet g.f.: zeta(s) * (1 - 1/2^(s-1)) * (1 - 1/3^s). - Amiram Eldar, Jun 09 2025