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%I #15 Jan 28 2024 02:00:57
%S 0,1,2,-1,0,1,-2,-1,0,4,2,-1,0,1,-2,-1,0,1,8,-1,0,1,-2,-1,0,1,2,-4,0,
%T 1,-2,-1,0,1,2,-1,0,1,-2,-1,0,1,2,-1,0,4,-2,-1,0,1,2,-1,0,1,-8,-1,0,1,
%U 2,-1,0,1,-2,-4,0,1,2,-1,0,1,-2,-1,0,1,2,-1,0
%N Expansion of f(x) + 2*f(x^2) + 3*f(x^9) + 6*f(x^18) in powers of x where f(x) := x / (1 + x^2).
%F a(n) is multiplicative with a(2) = 1, a(2^e) = 0 if e>1, a(3) = -1, a(3^e) = 4 * (-1)^e if e>1, a(p^e) = 1 if p == 1 (mod 4), a(p^e) = (-1)^e if p == 3 (mod 4).
%F a(n) = -a(-n) = a(n+72) = a(n+36)*(-1)^(mod(n,4)=2) for all n in Z.
%F 0 = a(n) + a(n+18) + a(n+36) + a(n+54) for all n in Z.
%F Sum_{d|n} a(d) * (-1)^(n+d) = A258256(n) if n>0.
%F Sum_{k=1..n} abs(a(k)) ~ (4/3) * n. - _Amiram Eldar_, Jan 28 2024
%e G.f. = x + 2*x^2 - x^3 + x^5 - 2*x^6 - x^7 + 4*x^9 + 2*x^10 - x^11 + ...
%t a[ n_] := {1, 2, -1, 0}[[Mod[n, 4, 1]]] If[ Divisible[ n, 9], 4, 1] (-1)^Boole[Mod[n, 8] == 6];
%t a[ n_] := With[ {m = Mod[n, 72], f = #/(1 + #^2) &}, SeriesCoefficient[ f[x] + 2 f[x^2] + 3 f[x^9] + 6 f[x^18], {x, 0, m}]];
%o (PARI) {a(n) = [0, 1, 2, -1][n%4 + 1] * if(n%9, 1, 4) * (-1)^(n%8==6)};
%Y Cf. A258256.
%K sign,easy,mult
%O 0,3
%A _Michael Somos_, May 24 2015
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