A008653 Theta series of direct sum of 2 copies of hexagonal lattice.

1, 12, 36, 12, 84, 72, 36, 96, 180, 12, 216, 144, 84, 168, 288, 72, 372, 216, 36, 240, 504, 96, 432, 288, 180, 372, 504, 12, 672, 360, 216, 384, 756, 144, 648, 576, 84, 456, 720, 168, 1080, 504, 288, 528, 1008, 72, 864, 576, 372, 684, 1116, 216, 1176, 648, 36

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.

Convolution square of A004016.

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Denoted by E_{2,3}^{i\infinity}(\tau) in Kaneko and Sakai 2012 on page 7. - Michael Somos, Dec 27 2014

Examples

			G.f. = 1 + 12*q + 36*q^2 + 12*q^3 + 84*q^4 + 72*q^5 + 36*q^6 + 96*q^7 + ...
		

References

• Bruce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 460, Entry 3(i).
• J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, 1999, p. 110.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Michael Gilleland, Some Self-Similar Integer Sequences.
• Masanobu Kaneko and Yuichi Sakai, The Ramanujan-Serre Differential Operators and certain Elliptic Curves, arXiv:1201.1685 [math.NT], 2012.
• Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]
• Gabriele Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2.

Crossrefs

Cf. A004016, A046913, A005882, A005928.

Programs

Magma

Basis( ModularForms( Gamma0(3), 2), 70)[1]; /* Michael Somos, Jun 12 2014 */

Mathematica

a[ n_] := SeriesCoefficient[ ((QPochhammer[ q]^3 + 9 q QPochhammer[ q^9]^3) / QPochhammer[ q^3])^2, {q, 0, n}]; (* Michael Somos, May 26 2014 *)
a[ n_] := If[ n < 1, Boole[ n == 0], 12 Sum[ If[ Mod[ d, 3] > 0, d, 0], {d, Divisors @ n }]]; (* Michael Somos, May 26 2014 *)

PARI

{a(n) = if( n<1, n==0, 12 * (sigma(3*n) - 3*sigma(n)))}; /* Michael Somos, Jul 19 2004 */

PARI

{a(n) = if( n<0, 0, polcoeff( sum(k=1, n, 6 * x^k / (1 + x^k + x^(2*k)), 1 + x * O(x^n))^2, n))}; /* Michael Somos, Jul 19 2004 */

Sage

ModularForms( Gamma0(3), 2, prec=70).0; # Michael Somos, Jun 12 2014

Formula

Expansion of (theta_3(z)*theta_3(3z)+theta_2(z)*theta_2(3z))^2.

Expansion of a(q)^2 in powers of q where a() is a cubic AGM theta function.

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 + 9*v^2 + 16*w^2 - 6*u*v + 4*u*w - 24*v*w. - Michael Somos, Jul 19 2004

G.f.: 1 + 12* Sum_{k>0} x^k / (1 - x^k)^2 - 36* Sum_{k>0} x^(3*k) / (1 - x^(3*k))^2. - Michael Somos, Apr 15 2007

a(n) = 12 * A046913(n) unless n=0.

Sum_{k=1..n} a(k) ~ c * n^2, where c = 2*Pi^2/3. - Amiram Eldar, Jan 21 2024