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

%I A033685 #48 Feb 16 2025 08:32:36
%S A033685 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,
%T A033685 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,
%U A033685 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
%N A033685 Theta series of hexagonal lattice A_2 with respect to deep hole.
%C A033685 The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
%C A033685 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%C A033685 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A033685 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 111.
%H A033685 G. C. Greubel, <a href="/A033685/b033685.txt">Table of n, a(n) for n = 0..5000</a>
%H A033685 Gabriele Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A2.html">Home page for hexagonal (or triangular) lattice A2</a>.
%H A033685 N. J. A. Sloane, <a href="http://dx.doi.org/10.1063/1.527472">Theta series and magic numbers for diamond and certain ionic crystal structures</a>, J. Math. Phys. 28 (1987), 1653-1657.
%H A033685 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>.
%H A033685 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A033685 a(3*n) = a(3*n + 2) = 0.
%F A033685 a(3*n + 1) = A005882(n) = 3 * A033687(n) = -A005928(3*n + 1) = A004016(3*n + 1) / 2.
%F A033685 Expansion of 3 * eta(q^3)^3 / eta(q) in powers of q^(1/3).
%F A033685 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
%F A033685 Expansion of c(x^3) in powers of x where c(x) is a cubic AGM theta function. - _Michael Somos_, Oct 17 2006
%F A033685 From _Michael Somos_, Dec 25 2011: (Start)
%F A033685 G.f.: Sum_{i, j in Z} x^(3 * (i^2 + i*j + j^2 + i + j) + 1).
%F A033685 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)
%F A033685 a(n) = A217219(n)/2. - _N. J. A. Sloane_, Oct 05 2012
%F A033685 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
%F A033685 From _Amiram Eldar_, Oct 13 2022: (Start)
%F A033685 a(n) = 3*A045833(n).
%F A033685 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/(3*sqrt(3)) = 1.209199... (A248897). (End)
%e A033685 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 + ...
%e A033685 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) + ...
%t A033685 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 *)
%t A033685 s = 3q*(QPochhammer[q^9]^3/QPochhammer[q^3])+O[q]^100; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 09 2015 *)
%o A033685 (PARI) {a(n) = if( (n<0) || (n%3 != 1), 0, 3 * sumdiv( n, d, kronecker( d, 3)))}; \\ _Michael Somos_, Jul 16 2005
%o A033685 (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
%Y A033685 Cf. A004016, A005882, A005928, A033687, A045833, A217219, A248897.
%Y A033685 Cf. A000122, A000700, A010054, A121373.
%K A033685 nonn
%O A033685 0,2
%A A033685 _N. J. A. Sloane_