A033685 Theta series of hexagonal lattice A_2 with respect to deep hole.

0, 3, 0, 0, 3, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 3, 0, 0, 6, 0, 0, 0, 0, 0, 3, 0, 0, 6, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 9, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 3, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 6, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 6, 0, 0, 3, 0, 0, 6, 0

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

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).

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 3*x + 3*x^4 + 6*x^7 + 6*x^13 + 3*x^16 + 6*x^19 + 3*x^25 + 6*x^28 + ...
G.f. = 3*q^(1/3) + 3*q^(4/3) + 6*q^(7/3) + 6*q^(13/3) + 3*q^(16/3) + 6*q^(19/3) + ...
		

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 111.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Gabriele Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2.
• N. J. A. Sloane, Theta series and magic numbers for diamond and certain ionic crystal structures, J. Math. Phys. 28 (1987), 1653-1657.
• Michael Somos, Introduction to Ramanujan theta functions.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Crossrefs

Cf. A004016, A005882, A005928, A033687, A045833, A217219, A248897.

Cf. A000122, A000700, A010054, A121373.

Programs

Mathematica

a[n_] := If[Mod[n, 3] != 1, 0, 3*DivisorSum[n, KroneckerSymbol[#, 3]&]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 03 2015, adapted from PARI *)
s = 3q*(QPochhammer[q^9]^3/QPochhammer[q^3])+O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 09 2015 *)

PARI

{a(n) = if( (n<0) || (n%3 != 1), 0, 3 * sumdiv( n, d, kronecker( d, 3)))}; \\ Michael Somos, Jul 16 2005

PARI

{a(n) = my(A); if( (n<0) || (n%3 != 1), 0, n = n\3; A = x * O(x^n); 3 * polcoeff( eta(x^3 + A)^3 / eta(x + A), n))}; \\ Michael Somos, Jul 16 2005

Formula

a(3*n) = a(3*n + 2) = 0.

a(3*n + 1) = A005882(n) = 3 * A033687(n) = -A005928(3*n + 1) = A004016(3*n + 1) / 2.

Expansion of 3 * eta(q^3)^3 / eta(q) in powers of q^(1/3).

G.f.: 3 * x * Product_{k>0} (1 - x^(9*k))^3 / (1 - x^(3*k)) = 3 * Sum_{k>0} x^k * (1 - x^k) * (1 - x^(2*k)) * (1 - x^(4*k)) / (1 - x^(9*k)). - Michael Somos, Jul 15 2005

Expansion of c(x^3) in powers of x where c(x) is a cubic AGM theta function. - Michael Somos, Oct 17 2006

From Michael Somos, Dec 25 2011: (Start)

G.f.: Sum_{i, j in Z} x^(3 * (i^2 + i*j + j^2 + i + j) + 1).

G.f.: Sum_{i, j, k} x^(3 * Q(i, j, k) - 2) where Q(i, j, k) = i*i + j*j + k*k + i*j + i*k + j*k and the sum is over all integer i, j, k where i + j + k = 1. (End)

a(n) = A217219(n)/2. - N. J. A. Sloane, Oct 05 2012

Expansion of 2 * x * psi(x^6) * f(x^6, x^12) + x * phi(x^3) * f(x^3, x^15) in powers of x where phi(), psi() are Ramanujan theta functions and f(, ) is Ramanujan's general theta function. - Michael Somos, Sep 09 2018

From Amiram Eldar, Oct 13 2022: (Start)

a(n) = 3*A045833(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/(3*sqrt(3)) = 1.209199... (A248897). (End)