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A008653
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Theta series of direct sum of 2 copies of hexagonal lattice.
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9
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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
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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.
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
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REFERENCES
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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.
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LINKS
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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.
Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
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FORMULA
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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.
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EXAMPLE
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G.f. = 1 + 12*q + 36*q^2 + 12*q^3 + 84*q^4 + 72*q^5 + 36*q^6 + 96*q^7 + ...
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MATHEMATICA
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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 *)
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PROG
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(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 */
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CROSSREFS
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Cf. A004016, A046913.
Sequence in context: A009649 A195539 A007794 * A038006 A205967 A203378
Adjacent sequences: A008650 A008651 A008652 * A008654 A008655 A008656
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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