A217221 - OEIS

%I #23 May 30 2026 16:40:38

%S 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,

%T 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,

%U 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

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

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

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

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

%H Antti Karttunen, <a href="/A217221/b217221.txt">Table of n, a(n) for n = 0..65537</a>

%H N. J. A. Sloane, <a href="https://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 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

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

%F Expansion of a(q) - a(q^4) in powers of q where a() is a cubic AGM function. - _Michael Somos_, Feb 01 2017

%F 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

%F 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

%F 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

%F a(2*n) = 0. a(2*n + 1) = 6 * A033762(n). - _Michael Somos_, Feb 01 2017

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

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

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

%o (Magma) A := Basis( ModularForms( Gamma1(12), 1), 80); 6*A[2] + 6*A[4]; /* _Michael Somos_, Feb 01 2017 */

%Y Cf. A033762, A115978, A217220.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Oct 05 2012