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

0,2

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).

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

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

Example

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

Mathematica

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

Prog

(PARI) {a(n) = if( n<1 || n%2==0, 0, 6 * sumdiv(n, d, kronecker(-3, d)))}; /* Michael Somos, Feb 01 2017 */
(Magma) A := Basis( ModularForms( Gamma1(12), 1), 80); 6*A[2] + 6*A[4]; /* Michael Somos, Feb 01 2017 */

Crossrefs

Cf. A033762, A115978, A217220.

Sequence in context: A335614 A019110 A280508 * A158090 A010677 A021169

Adjacent sequences: A217218 A217219 A217220 * A217222 A217223 A217224

Keyword

nonn

Author

N. J. A. Sloane, Oct 05 2012

Status

approved