A035019 - OEIS

%I #25 Nov 06 2023 07:15:13

%S 1,6,6,6,12,6,6,12,6,12,12,6,6,12,12,6,12,12,12,6,18,12,12,12,12,6,12,

%T 12,6,12,12,6,12,24,12,12,6,12,6,12,12,12,12,6,12,12,12,24,12,6,18,12,

%U 12,12,12,12,18,12,12,12,12,12,12,6,12,18,12,12,12,12

%N Sizes of successive shells in hexagonal (or A_2) lattice.

%C The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.

%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 111.

%H T. D. Noe, <a href="/A035019/b035019.txt">Table of n, a(n) for n = 0..10000</a>

%H G. 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>

%F Nonzero coefficients in expansion of theta_3(q)*theta_3(q^3) + theta_2(q)*theta_2(q^3).

%F The corresponding powers of q are A003136. - _Robert Israel_, Jul 29 2016

%p S:=series(JacobiTheta2(0,q)*JacobiTheta2(0,q^3)+JacobiTheta3(0,q)*JacobiTheta3(0,q^3),q,1001):

%p subs(0=NULL,[seq(coeff(S,q,j),j=0..1000)]); # _Robert Israel_, Jul 29 2016

%t s = EllipticTheta[2, 0, q]*EllipticTheta[2, 0, q^3] + EllipticTheta[3, 0, q]* EllipticTheta[3, 0, q^3] + O[q]^1000; CoefficientList[s, q] /. 0 -> Nothing (* _Jean-François Alcover_, Sep 19 2016, after _Robert Israel_ *)

%Y Cf. A003136, A004016, A038590 (partial sums), A357112.

%K nonn,easy,nice

%O 0,2

%A _N. J. A. Sloane_