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%I #27 Nov 19 2023 21:16:39
%S 0,1,0,1,-1,1,0,1,0,1,0,1,-1,1,0,1,0,1,0,1,-1,1,0,1,0,1,0,1,-1,1,0,1,
%T 0,1,0,1,-1,1,0,1,0,1,0,1,-1,1,0,1,0,1,0,1,-1,1,0,1,0,1,0,1,-1,1,0,1,
%U 0,1,0,1,-1,1,0,1,0,1,0,1,-1,1,0,1,0,1,0,1
%N Period 8 sequence [0, 1, 0, 1, -1, 1, 0, 1, ...].
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,1).
%F Euler transform of length 10 sequence [0, 1, -1, 0, 1, 0, 0, 1, 0, -1].
%F Moebius transform is length 8 sequence [1, -1, 0, -1, 0, 0, 0, 1].
%F a(n) is multiplicative with a(2) = 0, a(4) = -1, a(2^e) = 0 if e > 2, a(p^e) = 1 if p > 2.
%F G.f.: x / (1 - x^2) - x^4 / (1 - x^8).
%F G.f.: (x + x^3 - x^4 + x^5 + x^7) / (1 - x^8).
%F G.f.: x * (1 - x^2) * (1 - x^5) * (1 - x^8) / ((1 - x^3) * (1 - x^10)).
%F G.f.: x / (1 - x) - x^2 / (1 - x^2) - x^4 / (1 - x^4) + x^8 / (1 - x^8).
%F G.f. A(x) = Sum_{k>0} F(x^k) where P(x) = x - x^2 - x^4 + x^8 = x * (1 - x^3) * (1 - x - x^4).
%F a(n) = a(-n) = a(n+8) for all n in Z.
%F a(n) = -(-1)^n * A259044(n).
%F a(2*n + 1) = 1. a(4*n + 2) = 0. a(8*n) = 0. a(8*n + 4) = -1.
%F Dirichlet g.f.: zeta(s)*(2^s-1)^2*(2^s+1)/8^s. - _Amiram Eldar_, Dec 29 2022
%e G.f. = x + x^3 - x^4 + x^5 + x^7 + x^9 + x^11 - x^12 + x^13 + x^15 + ...
%t a[ n_] := Mod[n, 2] - Boole[ Mod[n, 8] == 4];
%t a[ n_] := {0, 1, 0, 1, -1, 1, 0, 1}[[Mod[n, 8] + 1]];
%t a[ n_] := SeriesCoefficient[ x / (1 - x^2) - x^4 / (1 - x^8), {x, 0, Abs@n}];
%t PadRight[{},120,{0,1,0,1,-1,1,0,1}] (* _Harvey P. Dale_, May 11 2019 *)
%o (PARI) {a(n) = n%2 - (n%8==4)};
%o (PARI) {a(n) = [0, 1, 0, 1, -1, 1, 0, 1][n%8 + 1]};
%o (PARI) {a(n) = 1 - (n%2==0) - (n%4==0) + (n%8==0)};
%o (PARI) {a(n) = polcoeff( x / (1 - x^2) - x^4 / (1 - x^8) + x * O(x^abs(n)), abs(n))};
%o (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1 +x^2-x^3+x^4+x^6)/(1-x^8))); // _G. C. Greubel_, Jul 29 2018
%Y Cf. A259044.
%K sign,mult,easy
%O 0,1
%A _Michael Somos_, Dec 29 2016
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