cp's OEIS Frontend

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

%I A005881 M0187 #33 Aug 12 2017 03:35:44
%S A005881 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,
%T A005881 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,
%U A005881 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
%N A005881 Theta series of planar hexagonal lattice (A2) with respect to edge.
%C A005881 Also number of ways of writing n as the sum of a triangular number and three times a triangular number.
%C A005881 The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
%C A005881 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.
%C A005881 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%D A005881 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005881 Antti Karttunen, <a href="/A005881/b005881.txt">Table of n, a(n) for n = 0..10000</a>
%H A005881 J. H. Conway and N. J. A. Sloane, <a href="http://dx.doi.org/10.1007/978-1-4757-2016-7">Sphere Packings, Lattices and Groups</a>, Springer-Verlag, p. 103. see Equ. (13).
%H A005881 M. D. Hirschhorn, <a href="http://dx.doi.org/10.1016/j.disc.2004.08.045">The number of representations of a number by various forms</a>, Discrete Mathematics 298 (2005), 205-211.
%H A005881 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>
%H A005881 N. J. A. Sloane, <a href="http://dx.doi.org/10.1063/1.527472">Theta series and magic numbers for diamond and certain ionic crystal structures</a>, J. Math. Phys. 28 (1987), 1653-1657.
%H A005881 N. J. A. Sloane and B. K. Teo, <a href="http://dx.doi.org/10.1063/1.449551">Theta series and magic numbers for close-packed spherical clusters</a>, J. Chem. Phys. 83 (1985) 6520-6534.
%F A005881 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
%F A005881 a(n) = 2*A033762(n).
%p A005881 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)];
%t A005881 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 *)
%o A005881 (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 */
%o A005881 (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 */
%Y A005881 Cf. A033762.
%K A005881 nonn
%O A005881 0,1
%A A005881 _N. J. A. Sloane_