A005881 Theta series of planar hexagonal lattice (A2) with respect to edge. (Formerly M0187)

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

0,1

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.

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).

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 *)

Prog

(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 */

Crossrefs

Cf. A033762.

Sequence in context: A286123 A253243 A201396 * A218875 A218869 A144458

Adjacent sequences: A005878 A005879 A005880 * A005882 A005883 A005884

Keyword

nonn

Author

N. J. A. Sloane

Status

approved