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 A217220 Theta series of Kagome net with respect to an atom. 2
 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, 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, 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 (list; graph; refs; listen; history; text; internal format)
 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), pp. 1653-1657. Michael Somos, Introduction to Ramanujan theta functions Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Phi_0(q)-phi_1(q^4) in the notation of SPLAG, Chapter 4. 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 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 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 EXAMPLE G.f. = 1 + 4*q + 4*q^3 + 6*q^4 + 8*q^7 + 4*q^9 + 6*q^12 + 8*q^13 + ... MAPLE S:= series(JacobiTheta3(0, q)*JacobiTheta3(0, q^3)+JacobiTheta2(0, q)*JacobiTheta2(0, q^3)/2, q, 103): seq(coeff(S, q, n), n=0..102); # Robert Israel, Nov 20 2017 MATHEMATICA 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 *) PROG (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 */ (Magma) A := Basis( ModularForms( Gamma1(12), 1), 80); A[1] + 4*A[2] + 4*A[4] + 6*A[5]; /* Michael Somos, Feb 01 2017 */ CROSSREFS Cf. A217221. Sequence in context: A258860 A134944 A088375 * A361620 A303560 A341481 Adjacent sequences: A217217 A217218 A217219 * A217221 A217222 A217223 KEYWORD nonn AUTHOR N. J. A. Sloane, Oct 05 2012 STATUS approved

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Last modified September 13 05:16 EDT 2024. Contains 375859 sequences. (Running on oeis4.)