cp's OEIS Frontend

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

%I A131729 #43 Dec 12 2023 09:26:42
%S A131729 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,
%T A131729 -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,
%U A131729 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
%N A131729 Period 4: repeat [0, 1, -1, 1].
%H A131729 Michael Somos, <a href="http://cis.csuohio.edu/~somos/rfmc.txt">Rational Function Multiplicative Coefficients</a>
%H A131729 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,1).
%F A131729 From _Michael Somos_, Apr 10 2011: (Start)
%F A131729 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.
%F A131729 Euler transform of length 6 sequence [-1, 1, 1, 1, 0, -1].
%F A131729 Moebius transform is length 4 sequence [1, -2, 0, 1].
%F A131729 a(n) is multiplicative with a(2) = -1, a(2^e) = 0 if e>1, a(p^e)=1 if p>2.
%F A131729 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)
%F A131729 G.f.: A081360 = Product_{k>0} (1 - x^k)^a(k). - _Michael Somos_, Feb 06 2012
%F A131729 G.f.: x*(1-x+x^2)/ ((1-x)*(x+1)*(x^2+1)). - _R. J. Mathar_, Nov 15 2007
%F A131729 a(n) = 1/4+(1/2)*cos(1/2*Pi*n)+3/4*(-1)^(1+n). - _R. J. Mathar_, Nov 15 2007
%F A131729 Dirichlet g.f. (1-2^(-s))^2*zeta(s). - _R. J. Mathar_, Apr 14 2011
%e A131729 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 + ...
%p A131729 seq(op([0, 1, -1, 1]), n=0..40); # _Wesley Ivan Hurt_, Jul 09 2016
%t A131729 a[ n_] := {1, -1, 1, 0}[[Mod[n, 4, 1]]]; (* _Michael Somos_, Nov 11 2015 *)
%o A131729 (PARI) a(n)=!(n%4)-(-1)^n \\ _Jaume Oliver Lafont_, Aug 28 2009
%o A131729 (PARI) {a(n) = [ 0, 1, -1, 1][n%4 + 1]}; /* _Michael Somos_, Apr 10 2011 */
%o A131729 (Magma) &cat [[0, 1, -1, 1]^^30]; // _Wesley Ivan Hurt_, Jul 09 2016
%Y A131729 Cf. A081360, A209635 (Dirichlet inverse), A166486 (absolute values).
%K A131729 sign,mult,easy
%O A131729 0,1
%A A131729 _Paul Curtz_, Sep 17 2007