A131729 Period 4: repeat [0, 1, -1, 1].

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

Offset

0,1

Links

Table of n, a(n) for n=0..113.

Michael Somos, Rational Function Multiplicative Coefficients

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

Formula

From Michael Somos, Apr 10 2011: (Start)

Expansion of x * (1 - x) * (1 - x^6) / ((1 - x^2) * (1 - x^3) * (1 - x^4)) = (x - x^2 + x^3) / (1 - x^4) in powers of x.

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

Moebius transform is length 4 sequence [1, -2, 0, 1].

a(n) is multiplicative with a(2) = -1, a(2^e) = 0 if e>1, a(p^e)=1 if p>2.

E.g.f.: sinh(x) + (cos(x) - cosh(x)) / 2. a(n) = a(-n) = a(n+4) for all n in Z. a(2*n + 1) = 0. a(4*n + 2) = -1. a(4*n) = 0. (End)

G.f.: A081360 = Product_{k>0} (1 - x^k)^a(k). - Michael Somos, Feb 06 2012

G.f.: x*(1-x+x^2)/ ((1-x)*(x+1)*(x^2+1)). - R. J. Mathar, Nov 15 2007

a(n) = 1/4+(1/2)*cos(1/2*Pi*n)+3/4*(-1)^(1+n). - R. J. Mathar, Nov 15 2007

Dirichlet g.f. (1-2^(-s))^2*zeta(s). - R. J. Mathar, Apr 14 2011

Example

G.f. = x - x^2 + x^3 + x^5 - x^6 + x^7 + x^9 - x^10 + x^11 + x^13 - x^14 + x^15 + ...

Maple

seq(op([0, 1, -1, 1]), n=0..40); # Wesley Ivan Hurt, Jul 09 2016

Mathematica

a[ n_] := {1, -1, 1, 0}[[Mod[n, 4, 1]]]; (* Michael Somos, Nov 11 2015 *)

Prog

(PARI) a(n)=!(n%4)-(-1)^n \\ Jaume Oliver Lafont, Aug 28 2009
(PARI) {a(n) = [ 0, 1, -1, 1][n%4 + 1]}; /* Michael Somos, Apr 10 2011 */
(Magma) &cat [[0, 1, -1, 1]^^30]; // Wesley Ivan Hurt, Jul 09 2016

Crossrefs

Cf. A081360, A209635 (Dirichlet inverse), A166486 (absolute values).

Sequence in context: A166486 A046978 A075553 * A144609 A115517 A351039

Adjacent sequences: A131726 A131727 A131728 * A131730 A131731 A131732

Keyword

sign,mult,easy

Author

Paul Curtz, Sep 17 2007

Status

approved