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
Examples
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 + ...
Links
- Michael Somos, Rational Function Multiplicative Coefficients
- Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).
Programs
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Magma
&cat [[0, 1, -1, 1]^^30]; // Wesley Ivan Hurt, Jul 09 2016
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Maple
seq(op([0, 1, -1, 1]), n=0..40); # Wesley Ivan Hurt, Jul 09 2016
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Mathematica
a[ n_] := {1, -1, 1, 0}[[Mod[n, 4, 1]]]; (* Michael Somos, Nov 11 2015 *) -
PARI
a(n)=!(n%4)-(-1)^n \\ Jaume Oliver Lafont, Aug 28 2009
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PARI
{a(n) = [ 0, 1, -1, 1][n%4 + 1]}; /* Michael Somos, Apr 10 2011 */
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