A034933 Expansion of theta_3(q)^2 * theta_3(q^3) in powers of q.

1, 4, 4, 2, 12, 16, 0, 8, 20, 4, 8, 8, 10, 32, 8, 0, 28, 24, 4, 8, 32, 16, 16, 16, 0, 28, 8, 2, 40, 48, 8, 8, 52, 0, 8, 16, 12, 64, 16, 8, 40, 24, 0, 24, 40, 16, 16, 16, 26, 28, 20, 0, 64, 80, 0, 16, 40, 24, 24, 8, 0, 64, 24, 8, 60, 48, 8, 24, 72, 0, 16, 16, 20, 48, 24, 10, 40, 96

Offset

0,2

Comments

a(n)=0 if n == 6*9^k (mod 9^(k+1)) for some k>=0. - Robert Israel, Aug 11 2019

Links

Robert Israel, Table of n, a(n) for n = 0..10000

Formula

Number of integer solutions to x^2 + y^2 + 3*z^2 = n.

Euler transform of period 12 sequence [4, -6, 6, -2, 4, -9, 4, -2, 6, -6, 4, -3, ...]. - Michael Somos, Sep 21 2005

Expansion of (eta(q^2)^2 * eta(q^6))^5 / (eta(q)^2 * eta(q^3) * eta(q^4)^2 * eta(q^12))^2 in power of q. - Michael Somos, Sep 21 2005

G.f.: theta_3(q)^2 * theta_3(q^3).

Example

1 + 4*q + 4*q^2 + 2*q^3 + 12*q^4 + 16*q^5 + 8*q^7 + 20*q^8 + 4*q^9 +...

Maple

S:= series(JacobiTheta3(0, q)^2*JacobiTheta3(0, q^3), q, 101):
seq(coeff(S, q, i), i=0..100); # Robert Israel, Aug 11 2019

Mathematica

CoefficientList[EllipticTheta[3, 0, q]^2*EllipticTheta[3, 0, q^3]+O[q]^80, q] (* Jean-François Alcover, Nov 27 2015 *)

Prog

(PARI) {a(n) = if( n<1, n==0, qfrep( [ 1, 0, 0; 0, 1, 0; 0, 0, 3], n)[n] * 2)} /* Michael Somos, Sep 21 2005 */

Crossrefs

Sequence in context: A232523 A376878 A224821 * A320148 A320147 A369410

Adjacent sequences: A034930 A034931 A034932 * A034934 A034935 A034936

Keyword

nonn

Author

N. J. A. Sloane

Status

approved