A217221 Theta series of Kagome net with respect to a deep hole.

0, 6, 0, 6, 0, 0, 0, 12, 0, 6, 0, 0, 0, 12, 0, 0, 0, 0, 0, 12, 0, 12, 0, 0, 0, 6, 0, 6, 0, 0, 0, 12, 0, 0, 0, 0, 0, 12, 0, 12, 0, 0, 0, 12, 0, 0, 0, 0, 0, 18, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 12, 0, 12, 0, 0, 0, 12, 0, 0, 0, 0, 0, 12, 0, 6, 0, 0, 0, 12, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24, 0, 12, 0, 0, 0, 12, 0, 0, 0, 0, 0

Offset: 0

Author

N. J. A. Sloane, Oct 05 2012

Keywords

• nonn

Comments

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

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 6*q + 6*q^3 + 12*q^7 + 6*q^9 + 12*q^13 + 12*q^19 + 12*q^21 + ...
		

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag.

Links

• Antti Karttunen, Table of n, a(n) for n = 0..65537
• N. J. A. Sloane, Theta series and magic numbers for diamond and certain ionic crystal structures, J. Math. Phys. 28 (1987), 1653-1657.
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A033762, A115978, A217220.

Programs

Magma

A := Basis( ModularForms( Gamma1(12), 1), 80); 6*A[2] + 6*A[4]; /* Michael Somos, Feb 01 2017 */

Mathematica

a[ n_] := If[ n < 1 || EvenQ[n], 0, 6 DivisorSum[n, Mod[(3 - #)/2, 3, -1] &]]; (* Michael Somos, Feb 01 2017 *)

PARI

{a(n) = if( n<1 || n%2==0, 0, 6 * sumdiv(n, d, kronecker(-3, d)))}; /* Michael Somos, Feb 01 2017 */

Formula

Phi_0(q)-phi_0(q^4) in the notation of SPLAG, Chapter 4.

Expansion of a(q) - a(q^4) in powers of q where a() is a cubic AGM function. - Michael Somos, Feb 01 2017

Expansion of 6 * q * psi(q^2) * psi(q^6) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Feb 01 2017

Expansion of 6 * (eta(q^4) * eta(q^12))^2 / (eta(q^2) * eta(q^6)) in powers of q. - Michael Somos, Feb 01 2017

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 27^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A115978. - Michael Somos, Feb 01 2017

a(2*n) = 0. a(2*n + 1) = 6 * A033762(n). - Michael Somos, Feb 01 2017