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A131729 Period 4: repeat [0, 1, -1, 1]. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

LINKS

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

M. 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

a(n) = (1/24)*{7*(n mod 4)-11*[(n+1) mod 4]+13*[(n+2) mod 4]-5*[(n+3) mod 4]}. - Paolo P. Lava, Oct 02 2007

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

a(n) = 1/4+(1/4)*I^n-(3/4)*(-1)^n+(1/4)*(-I)^n. - Paolo P. Lava, Jul 17 2008

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 A022930

Adjacent sequences:  A131726 A131727 A131728 * A131730 A131731 A131732

KEYWORD

sign,mult,easy

AUTHOR

Paul Curtz, Sep 17 2007

STATUS

approved

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Last modified July 21 06:28 EDT 2019. Contains 325191 sequences. (Running on oeis4.)