

A005882


Theta series of planar hexagonal lattice (A2) with respect to deep hole.
(Formerly M2281)


203



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

0,1


COMMENTS

The hexagonal lattice is the familiar 2dimensional 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).
On page 111 of Conway and Sloane is "If the origin is moved to a deep hole the theta series is Theta_{hex+[1]}(z) = theta_2(z) psi_6(3z) + theta_3(z) psi_3(3z) = 3 q^{1/3} + 3 q^{4/3} + 6 q^{7/3} + 6 q^{13/3} + ... (63)" where the psi_k() for integer k is defined on page 103 equation (11) as psi_k(z) = e^{Pi i/z^2} theta_3(Pi z/k  z) = Sum_{m in Z} q^{(m + 1/k)^2}.  Michael Somos, Sep 10 2018


REFERENCES

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", SpringerVerlag, p. 111.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
J. M. Borwein and P. B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691701. MR1010408 (91e:33012) see page 695.
Christian Kassel and Christophe Reutenauer, The zeta function of the Hilbert scheme of n points on a twodimensional torus, arXiv:1505.07229v3 [math.AG], 2015. [Note that a later version of this paper has a different title and different contents, and the numbertheoretical part of the paper was moved to the publication which is next in this list.]
Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a twodimensional torus, arXiv:1610.07793 [math.NT], 2016.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
N. J. A. Sloane and B. K. Teo, Theta series and magic numbers for closepacked spherical clusters, J. Chem. Phys. 83 (1985) 65206534.
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions


FORMULA

Expansion of q^(1/3) * 3 * eta(q^3)^3 / eta(q) in powers of q.
Expansion of q^(1/3) * c(q) in powers of q where c(q) is the third cubic AGM theta function.
Given g.f. A(x), then B(x) = x*A(x^3) satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = v^3 + 2*u*w^2  u^2*w.  Michael Somos, Aug 15 2006
G.f.: 3 Product_{k>0} (1q^(3k))^3/(1q^k).
G.f.: Sum_{u,v in Z} x^(u*u + u*v + v*v + u + v).  Michael Somos, Jul 19 2014
a(n) = 3 * A033687(n). a(n) = A113062(3*n + 1) = A033685(3*n + 1).
Expansion of 2 * psi(x^2) * f(x^2, x^4) + phi(x) * f(x^1, x^5) in powers of x where phi(), psi() are Ramanujan theta functions and f(, ) is Ramanujan's general theta function.  Michael Somos, Sep 07 2018


EXAMPLE

G.f. = 3 + 3*x + 6*x^2 + 6*x^4 + 3*x^5 + 6*x^6 + 3*x^8 + 6*x^9 + 6*x^10 + ...
G.f. = 3*q + 3*q^4 + 6*q^7 + 6*q^13 + 3*q^16 + 6*q^19 + 3*q^25 + 6*q^28 + ...


MATHEMATICA

a[ n_] := SeriesCoefficient[ 3 QPochhammer[ q^3]^3 / QPochhammer[ q], {q, 0, n}]; (* Michael Somos, Jul 19 2014 *)


PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( 3 * eta(x^3 + A)^3 / eta(x + A), n))}; /* Michael Somos, Aug 15 2006 */
(MAGMA) Basis( ModularForms( Gamma1(9), 1), 302)[2] * 3; /* Michael Somos, Jul 19 2014 */


CROSSREFS

Essentially same as A033685 and A033687.
Cf. A113062, A273845.
Sequence in context: A132809 A265960 A279062 * A189915 A085572 A205548
Adjacent sequences: A005879 A005880 A005881 * A005883 A005884 A005885


KEYWORD

nonn


AUTHOR

N. J. A. Sloane


STATUS

approved



