A045866 Theta series of quadratic form with Gram matrix [ 2, 1, 0, 1; 1, 8, 1, -3; 0, 1, 10, 2; 1, -3, 2, 12 ].

1, 2, 0, 0, 6, 2, 14, 4, 10, 14, 16, 4, 12, 6, 18, 22, 18, 14, 14, 14, 24, 24, 34, 20, 40, 18, 32, 16, 36, 24, 36, 16, 50, 48, 36, 34, 74, 0, 40, 42, 60, 16, 58, 34, 44, 40, 44, 24, 96, 30, 64, 50, 64, 36, 98, 58, 80, 54, 48, 52, 124, 36, 72, 64, 74, 60, 66, 52, 80, 60, 92, 52

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

This is the 4-dimensional Elkies_B lattice.

Examples

			1 + 2*q^2 + 6*q^8 + 2*q^10 + 14*q^12 + 4*q^14 + 10*q^16 + 14*q^18 + 16*q^20 + ...
		

Links

• John Cannon, Table of n, a(n) for n = 0..5000
• N. D. Elkies, Elliptic and modular curves over finite fields and related computational issues, in AMS/IP Studies in Advanced Math., 7 (1998), 21-76, esp. p. 57.

Crossrefs

Dual lattice to that in A045867.

Programs

PARI

B(x, y, z, w)=2*x^2+8*y^2+10*z^2+12*w^2+2*x*(y+w)+2*y*(z-3*w)+4*z*w;
thetaB(n, N, bx, by, bz, bw, ix, iy, iz, iw, nn)=n=2*n; bx=floor(sqrt(n)*(1+1/sqrt(6))); bz=floor(sqrt(n/7)); bw=floor(sqrt(n/6)); by=floor(sqrt(n/3)); N=vector(n/2+2); for(ix=-bx, bx, for(iy=-by, by, for(iz=-bz, bz, for(iw=-bw, bw, nn=B(ix, iy, iz, iw); if (nn<=n, N[1+nn/2]++); )))); N;
thetaB(80)

PARI

{a(n)=if(n<1, n==0, qfrep([ 2, 1, 0, 1; 1, 8, 1, -3; 0, 1, 10, 2; 1, -3, 2, 12 ], n, 1)[n]*2)} /* Michael Somos, Apr 02 2006 */

Extensions

More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 22 2000