This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A217220 #21 Feb 16 2025 08:33:18
%S A217220 1,4,0,4,6,0,0,8,0,4,0,0,6,8,0,0,6,0,0,8,0,8,0,0,0,4,0,4,12,0,0,8,0,0,
%T A217220 0,0,6,8,0,8,0,0,0,8,0,0,0,0,6,12,0,0,12,0,0,0,0,8,0,0,0,8,0,8,6,0,0,
%U A217220 8,0,0,0,0,0,8,0,4,12,0,0,8,0,4,0,0,12,0,0,0,0,0,0,16,0,8,0,0,0,8,0,0,6,0,0
%N A217220 Theta series of Kagome net with respect to an atom.
%C A217220 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%C A217220 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A217220 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag.
%H A217220 Antti Karttunen, <a href="/A217220/b217220.txt">Table of n, a(n) for n = 0..65537</a>
%H A217220 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), pp. 1653-1657.
%H A217220 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A217220 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A217220 Phi_0(q)-phi_1(q^4) in the notation of SPLAG, Chapter 4.
%F A217220 a(n) = 4 * b(n) where b() is multiplicative with b(2^e) = (1+(-1)^e)*3/4, b(3^e) = 1, b(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6), b(p^e) = e+1 if p == 1 (mod 6). - _Michael Somos_, Feb 01 2017
%F A217220 Expansion of (2 * a(q) + a(q^4)) / 3 in powers of q where a() is a cubic AGM function. - _Michael Somos_, Feb 01 2017
%F A217220 Expansion of phi(q) * phi(q^3) + 2 * q * psi(q^2) * psi(q^6) in powers of q where phi(), psi() are Ramanujan theta functions. - _Michael Somos_, Feb 01 2017
%e A217220 G.f. = 1 + 4*q + 4*q^3 + 6*q^4 + 8*q^7 + 4*q^9 + 6*q^12 + 8*q^13 + ...
%p A217220 S:= series(JacobiTheta3(0,q)*JacobiTheta3(0,q^3)+JacobiTheta2(0,q)*JacobiTheta2(0,q^3)/2, q, 103):
%p A217220 seq(coeff(S,q,n),n=0..102); # _Robert Israel_, Nov 20 2017
%t A217220 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^3] + 1/2 EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^3], {q, 0, n}]; (* _Michael Somos_, Feb 01 2017 *)
%o A217220 (PARI) {a(n) = if( n<1, n==0, 4 * sumdiv( n, d, kronecker( d, 3)) + if( n%4==0, 2 * sumdiv( n/4, d, kronecker( d, 3))))}; /* _Michael Somos_, Feb 01 2017 */
%o A217220 (Magma) A := Basis( ModularForms( Gamma1(12), 1), 80); A[1] + 4*A[2] + 4*A[4] + 6*A[5]; /* _Michael Somos_, Feb 01 2017 */
%Y A217220 Cf. A217221.
%K A217220 nonn
%O A217220 0,2
%A A217220 _N. J. A. Sloane_, Oct 05 2012