A089806 Expansion of Jacobi theta function (theta_3(q^(1/3))-theta_2(q^3))/2/q^(1/12).

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, 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, 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

Offset

0,1

Links

Antti Karttunen, Table of n, a(n) for n = 0..10034

Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Eric Weisstein's World of Mathematics, Jacobi Theta Functions.

I. J. Zucker, Further Relations Amongst Infinite Series and Products. II. The Evaluation of Three-Dimensional Lattice Sums, J. Phys. A: Math. Gen. 23, 117-132, 1990.

Index entries for characteristic functions.

Formula

Euler transform of period 12 sequence [0, 1, 0, 0, 0, -1, 0, 0, 0, 1, 0, -1, ...]. - Michael Somos, Apr 13 2005

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

Expansion of q^(-1/12)(eta(q^4)eta(q^6)^2)/(eta(q^2)eta(q^12)) in powers of q.

Sum_{k=1..n} a(k) ~ c * sqrt(n), where c = 2/sqrt(3) = 1.1547005... (10 * A020832). - Amiram Eldar, Dec 29 2023

Example

1 + q^2 + q^4 + q^10 + q^14 + q^24 + q^30 + q^44 + q^52 + ...

Mathematica

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 *)
Table[If[IntegerQ[Sqrt[12*n + 1]], 1, 0], {n, 0, 100}] (* Vaclav Kotesovec, Dec 29 2023 *)

Prog

(PARI) a(n)=issquare(12*n+1) /* Michael Somos, Apr 13 2005 */
(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

Crossrefs

Cf. A080995(n) = a(2n).

Cf. A020832.

Sequence in context: A014141 A014093 A373601 * A274719 A014069 A154388

Adjacent sequences: A089803 A089804 A089805 * A089807 A089808 A089809

Keyword

nonn,easy,changed

Author

Eric W. Weisstein, Nov 12 2003

Status

approved