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A244612 a(n) = 1 if n is a square, -1 if n is six times a square, 0 if n < 1.

%I A244612 #17 Feb 16 2025 08:33:23
%S A244612 1,0,0,1,0,-1,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,
%T A244612 0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,
%U A244612 0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,0,0
%N A244612 a(n) = 1 if n is a square, -1 if n is six times a square, 0 if n < 1.
%C A244612 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A244612 Antti Karttunen, <a href="/A244612/b244612.txt">Table of n, a(n) for n = 1..65537</a>
%H A244612 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>.
%H A244612 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A244612 Expansion of (phi(q) - phi(q^6)) / 2 in powers of q where phi() is a Ramanujan theta function.
%F A244612 G.f.: (theta_3(q) - theta_3(q^6)) / 2 = Sum_{k>0} x^(k^2) - x^(6*k^2).
%F A244612 a(3*n) = A089801(n). a(3*n + 2) = 0.
%F A244612 Sum_{k=1..n} a(k) ~ c*sqrt(n), where c = 1 - 1/sqrt(6) = 0.5917517... . - _Amiram Eldar_, Oct 24 2023
%e A244612 G.f. = q + q^4 - q^6 + q^9 + q^16 - q^24 + q^25 + q^36 + q^49 - q^54 + ...
%t A244612 a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] - EllipticTheta[ 3, 0, q^6]) / 2,  {q, 0, n}];
%t A244612 a[ n_] := If[ n < 0, 0, Boole[ OddQ [ Length @ Divisors @ n]] - Boole[ OddQ [ Length @ Divisors [6 n]]]];
%o A244612 (PARI) {a(n) = issquare(n) - issquare(6*n)};
%o A244612 (Magma) Basis( ModularForms( Gamma1(24), 1/2), 64) [2];
%Y A244612 Cf. A089801, A214293, A214295, A245485.
%K A244612 sign
%O A244612 1,1
%A A244612 _Michael Somos_, Jul 01 2014
%E A244612 More terms from _Antti Karttunen_, Dec 15 2017