A005878 Theta series of cubic lattice with respect to deep hole.

8, 24, 24, 32, 48, 24, 48, 72, 24, 56, 72, 48, 72, 72, 48, 48, 120, 72, 56, 96, 24, 120, 120, 48, 96, 96, 72, 96, 120, 48, 104, 168, 96, 48, 120, 72, 96, 192, 72, 144, 96, 72, 144, 120, 96, 104, 192, 72, 120, 192, 48, 144, 216, 48, 96, 120, 144, 192, 168, 120, 96, 216, 72

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Number of ways of writing 8*n+3 as the sum of three odd squares. - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 23 2008

Expansion of Jacobi theta constant theta_2^3. - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 23 2008

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 107.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..99 from Herman Jamke (hermanjamke(AT)fastmail.fm))
• G. Nebe and N. J. A. Sloane, Home page for this lattice

Crossrefs

Equals 8 times A008443. Cf. A085121.

Programs

Mathematica

QP = QPochhammer; CoefficientList[(2 QP[q^2]^2/QP[q])^3 + O[q]^63, q] (* Jean-François Alcover, Jul 04 2017 *)

PARI

{a(n)=if(n<0, 0, 8*polcoeff( sum(k=0, (sqrtint(8*n+1)-1)\2, x^((k^2+k)/2), x*O(x^n))^3, n))} {a(n)=local(A); if(n<0, 0, A=x*O(x^n); 8*polcoeff( (eta(x^2+A)^2/eta(x+A))^3, n))} \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 23 2008

Formula

G.f.: Form (Sum_{n=-inf..inf} q^((2n+1)^2))^3, then divide by q^3 and set q = x^(1/8).

a(n) = 8 * A008443(n).

Extensions

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 23 2008