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A033685
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Theta series of hexagonal lattice A_2 with respect to deep hole.
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9
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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
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OFFSET
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0,2
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COMMENTS
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The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
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REFERENCES
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J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 111.
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LINKS
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FORMULA
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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
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)
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
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/(3*sqrt(3)) = 1.209199... (A248897). (End)
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EXAMPLE
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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) + ...
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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