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%I #29 Dec 29 2023 04:08:09
%S 1,0,1,0,1,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,
%T 0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U 0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N Expansion of Jacobi theta function (theta_3(q^(1/3))-theta_2(q^3))/2/q^(1/12).
%H Antti Karttunen, <a href="/A089806/b089806.txt">Table of n, a(n) for n = 0..10034</a>
%H Vaclav Kotesovec, <a href="/A089806/a089806.jpg">Graph - the asymptotic ratio (100000 terms)</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>.
%H I. J. Zucker, <a href="https://doi.org/10.1088/0305-4470/23/2/009">Further Relations Amongst Infinite Series and Products. II. The Evaluation of Three-Dimensional Lattice Sums</a>, J. Phys. A: Math. Gen. 23, 117-132, 1990.
%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>.
%F Euler transform of period 12 sequence [0, 1, 0, 0, 0, -1, 0, 0, 0, 1, 0, -1, ...]. - _Michael Somos_, Apr 13 2005
%F a(n) = b(12n+1) where b(n) is multiplicative and b(3^e)=0^e, b(p^e)=(1+(-1)^e)/2 if p<>3. - _Michael Somos_, Jun 06 2005
%F Expansion of q^(-1/12)(eta(q^4)eta(q^6)^2)/(eta(q^2)eta(q^12)) in powers of q.
%F Sum_{k=1..n} a(k) ~ c * sqrt(n), where c = 2/sqrt(3) = 1.1547005... (10 * A020832). - _Amiram Eldar_, Dec 29 2023
%e 1 + q^2 + q^4 + q^10 + q^14 + q^24 + q^30 + q^44 + q^52 + ...
%t nmax = 100; CoefficientList[Series[Product[(1+x^(2*k)) * (1-x^(6*k)) / (1+x^(6*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jul 05 2016 *)
%t Table[If[IntegerQ[Sqrt[12*n + 1]], 1, 0], {n, 0, 100}] (* _Vaclav Kotesovec_, Dec 29 2023 *)
%o (PARI) a(n)=issquare(12*n+1) /* _Michael Somos_, Apr 13 2005 */
%o (PARI) lista(nn) = {q='q+O('q^nn); Vec(eta(q^4)*eta(q^6)^2/(eta(q^2)*eta(q^12)))} \\ _Altug Alkan_, Mar 22 2018
%Y Cf. A080995(n) = a(2n).
%Y Cf. A020832.
%K nonn,easy
%O 0,1
%A _Eric W. Weisstein_, Nov 12 2003
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