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A008653 Theta series of direct sum of 2 copies of hexagonal lattice. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

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

REFERENCES

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 460, Entry 3(i).

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 110.

Masao Koike, Modular forms on non-compact arithmetic triangle groups, preprint.

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Michael Gilleland, Some Self-Similar Integer Sequences

M. Kaneko and Y. Sakai, The Ramanujan-Serre Differential Operators and certain Elliptic Curves, arXiv:1201.1685 [math.NT], 2012.

G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2

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.

Convolution square of A004016.

EXAMPLE

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

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 *)

PROG

(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

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

CROSSREFS

Cf. A004016, A046913.

Sequence in context: A009649 A195539 A007794 * A038006 A205967 A203378

Adjacent sequences:  A008650 A008651 A008652 * A008654 A008655 A008656

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified January 21 13:55 EST 2020. Contains 331113 sequences. (Running on oeis4.)