A014453 Theta series of quadratic form with Gram matrix [ 2, 0, 0; 0, 2, 1; 0, 1, 2 ].

1, 8, 12, 6, 20, 24, 0, 24, 36, 8, 24, 24, 18, 48, 24, 0, 44, 48, 12, 24, 48, 24, 48, 48, 0, 56, 24, 6, 72, 72, 24, 24, 84, 0, 24, 48, 20, 96, 48, 24, 72, 48, 0, 72, 72, 24, 48, 48, 42, 56, 60, 0, 96, 120, 0, 48, 72, 48, 72, 24, 0, 96, 72, 24, 92, 96, 24, 72, 120, 0, 48, 48, 36, 96, 72

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

This is the hexagonal P lattice (the even holotype) of dimension 3.

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

a(n) is the number of solutions to x^2 + y^2 + z^2 + x*y = n in integers. - Michael Somos, Jul 03 2018

Examples

			G.f. = 1 + 8*x + 12*x^2 + 6*x^3 + 20*x^4 + 24*x^5 + 24*x^7 + 36*x^8 + 8*x^9 + ...
G.f. = 1 + 8*q^2 + 12*q^4 + 6*q^6 + 20*q^8 + 24*q^10 + 24*q^14 + 36*q^16 + 8*q^18 + ...
		

Links

• John Cannon, Table of n, a(n) for n = 0..5000
• G. Nebe and N. J. A. Sloane, Home page for this lattice
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A000122, A004016.

Programs

Mathematica

(* A004016 *) a2[0] = 1; a2[n_] := 6*DivisorSum[n, KroneckerSymbol[#, 3]&]; (* A000122 *) a3[n_] := SeriesCoefficient[EllipticTheta[3, 0, q], {q, 0, n}]; a[n_] := Sum[a2[k]*a3[n-k], {k, 0, n}]; Table[a[n], {n, 0, 75}] (* Jean-François Alcover, Nov 04 2015, from the convolution given by Michael Somos *)
a[ n_] :=   SeriesCoefficient[ EllipticTheta[ 3, 0,  x] (QPochhammer[ x]^3 + 9 x QPochhammer[ x^9]^3) / QPochhammer[ x^3], {x, 0, n}]; (* Michael Somos, Jul 03 2018 *)

PARI

{a(n) = if( n<1, n==0, 2 * qfrep( [ 2, 0, 0; 0, 2, 1; 0, 1, 2 ], n, 1)[n])}; /* Michael Somos, May 30 2012 */

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^3  + 9 * x * eta(x^9 + A)^3) / eta(x^3 + A) * eta(x^2 + A)^5 / (eta(x + A)^2 * eta(x^4 + A)^2), n))}; /* Michael Somos, May 30 2012 */

Formula

Expansion of a(x) * phi(x) where phi() is a Ramanujan theta function and a() is a cubic AGM theta function. - Michael Somos, May 30 2012

Expansion of (eta(q)^3 + 9 * eta(q^9)^3) * eta(q^2)^5 / (eta(q)^2 * eta(q^3) * eta(q^4)^2) in powers of q.

Convolution of A004016 and A000122. - Michael Somos, May 30 2012