OFFSET
0,1
COMMENTS
Period 6: repeat [0, 1, -1, 0, -1, 1]. - Joerg Arndt, Aug 28 2024
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
Michael Somos, Rational Function Multiplicative Coefficients.
Index entries for linear recurrences with constant coefficients, signature (0,0,-1).
FORMULA
a(n) = 2*cos(Pi*n/3)/3 - 2(-1)^n/3.
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) = 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
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 + ...
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 *)
PROG
(PARI) {a(n) = [0, 1, -1, 0, -1, 1][n%6 + 1]}; /* Michael Somos, Apr 10 2011 */
CROSSREFS
KEYWORD
sign,easy,mult
AUTHOR
Paul Barry, Feb 25 2004
STATUS
approved