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.
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.
FORMULA
a(3*n) = a(3*n + 2) = 0.
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)
EXAMPLE
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) + ...
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved