cp's OEIS Frontend

A008655 Theta series of direct sum of 4 copies of hexagonal lattice.

%I A008655 #23 Feb 22 2021 10:04:45
%S A008655 1,24,216,888,1752,3024,7992,8256,14040,24216,27216,31968,64824,52752,
%T A008655 74304,111888,112344,117936,217944,164640,220752,305472,287712,292032,
%U A008655 519480,378024,474768,654072
%N A008655 Theta series of direct sum of 4 copies of hexagonal lattice.
%C A008655 The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
%C A008655 Convolution of A008654 and A004016. Convolution square of A008653. - _R. J. Mathar_, Feb 22 2021
%D A008655 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 110.
%H A008655 Seiichi Manyama, <a href="/A008655/b008655.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from G. C. Greubel)
%H A008655 J. McKay and A. Sebbar, <a href="http://dx.doi.org/10.1007/s002080000116">Fuchsian groups, automorphic functions and Schwarzians</a>, Math. Ann., 318 (2000), 255-275.
%H A008655 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 A008655 Expansion of (theta_3(z)*theta_3(3z) + theta_2(z)*theta_2(3z))^4.
%p A008655 A008655 := proc(n)
%p A008655         add( A004016(i)*x^i,i=0..n) ;
%p A008655         coeftayl(%^4,x=0,n) ;
%p A008655 end proc: # _R. J. Mathar_, Feb 22 2021
%t A008655 terms = 28; s = (EllipticTheta[3, 0, q]^3 + EllipticTheta[3, Pi/3, q]^3 + EllipticTheta[3, 2 Pi/3, q]^3)^4/(81*EllipticTheta[3, 0, q^3]^4) + O[q]^(2 terms); CoefficientList[s, q^2][[1 ;; terms]] (* _Jean-François Alcover_, Jul 07 2017, from LatticeData(A2) *)
%K A008655 nonn,easy
%O A008655 0,2
%A A008655 _N. J. A. Sloane_