A005881 Theta series of planar hexagonal lattice (A2) with respect to edge.

2, 2, 0, 4, 2, 0, 4, 0, 0, 4, 4, 0, 2, 2, 0, 4, 0, 0, 4, 4, 0, 4, 0, 0, 6, 0, 0, 0, 4, 0, 4, 4, 0, 4, 0, 0, 4, 2, 0, 4, 2, 0, 0, 0, 0, 8, 4, 0, 4, 0, 0, 4, 0, 0, 4, 4, 0, 0, 4, 0, 2, 0, 0, 4, 4, 0, 8, 0, 0, 4, 0, 0, 0, 6, 0, 4, 0, 0, 4, 0, 0, 4, 0, 0, 6, 4, 0, 4, 0, 0, 4, 4, 0, 0, 4, 0, 4, 0, 0, 4, 4, 0, 0, 0, 0

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Also number of ways of writing n as the sum of a triangular number and three times a triangular number.

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

Given g.f. A(x), then q^(1/2)*A(q) is denoted phi_1(z) where q=exp(Pi*i*z) in Conway and Sloane.

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Antti Karttunen, Table of n, a(n) for n = 0..10000
• J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, p. 103. see Equ. (13).
• M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
• G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
• N. J. A. Sloane, Theta series and magic numbers for diamond and certain ionic crystal structures, J. Math. Phys. 28 (1987), 1653-1657.
• N. J. A. Sloane and B. K. Teo, Theta series and magic numbers for close-packed spherical clusters, J. Chem. Phys. 83 (1985) 6520-6534.

Crossrefs

Cf. A033762.

Programs

Maple

d:=proc(r,m,n) local i,t1; t1:=0; for i from 1 to n do if n mod i = 0 and i-r mod m = 0 then t1:=t1+1; fi; od: t1; end; [seq(2*(d(1,3,2*n+1)-d(2,3,2*n+1)),n=0..120)];

Mathematica

a[n_] := 2*DivisorSum[2n+1, KroneckerSymbol[-12, #]*Mod[(2n+1)/#, 2]& ]; Table[a[n], {n, 0, 105}] (* Jean-François Alcover, Dec 02 2015, adapted from PARI *)

PARI

{a(n) = if( n<0, 0, n = 2*n + 1; 2 * sumdiv(n, d, kronecker( -12, d) * (n/d%2)))}; /* Michael Somos, Nov 05 2006 */

PARI

{a(n) = if( n<0, 0, n = 8*n + 4; 2 * sum(j=1, sqrtint(n\3), (j%2) * issquare(n - 3*j^2)))}; /* Michael Somos, Nov 05 2006 */

Formula

Expansion of q^(-1) * (a(q) - a(q^4)) / 3 in powers of q^2 where a() is a cubic AGM theta function. - Michael Somos, Nov 05 2006

a(n) = 2*A033762(n).