A317965 Representation numbers (or theta series) for either one of Schiemann's pair of four-dimensional positive definite quadratic forms with the same representation numbers.

1, 0, 2, 0, 4, 6, 10, 6, 12, 6, 6, 8, 10, 8, 10, 22, 24, 4, 28, 12, 24, 20, 24, 14, 42, 20, 16, 14, 32, 10, 46, 8, 46, 30, 28, 28, 62, 34, 32, 40, 38, 28, 48, 28, 60, 50, 48, 32, 50, 28, 62, 34, 52, 26, 68, 30, 62, 56, 68, 38, 110, 28, 50, 64, 86, 60, 72, 50, 56, 34, 88, 50, 138

Offset

0,3

Comments

a(n) is the number of ways either form represents 2n.

a(n) is the number of integer solutions (x, y, z, w) to n = 2*x^2 + 4*y^2 + 5*z^2 + 5*w^2 + 2*x*y + 3*y*z + w*x + w*y + 5*w*z. The negative of a solution is a different solution unless n = 0. This implies that solutions come in pairs which implies a(n) is even unless n = 0. - Michael Somos, Apr 16 2022

References

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, Third edition, p. xxix.

Links

Ben Mares, Table of n, a(n) for n = 0..20000

J. H. Conway and N. J. A. Sloane, Four-Dimensional Lattices With the Same Theta Series, Duke Math. J., 66 (International Mathematics Research Notices Number 4) (1992), pp. 93-96. (Pages are in reverse order)

A. G. Earnest and G. Nipp, On the theta series of positive quaternary quadratic forms, CR Math. Rep. Acad. Sci. Canada 13 (1991): 33-38. See p. 36.

Alexander Schiemann, Ein Beispiel positiv definiter quadratischer Formen der Dimension 4 mit gleichen Darstellungszahlen, Archiv der Mathematik, 54.4 (1990): 372-375.

Example

G.f. = 1 + 2*x^2 + 4*x^4 + 6*x^5 + 10*x^6 + 6*x^7 + 12*x^8 + 6*x^9 + 6*x^10 + ... - Michael Somos, Apr 16 2022

Mathematica

a[ n_ ] := Module[{x, y, z, w}, Length @ FindInstance[ n == 2*x^2 + 4*y^2 + 5*z^2 + 5*w^2 + 2*x*y + 3*y*z + w*x + w*y + 5*w*z, {x, y, z, w}, Integers, 10^9]]; (* Michael Somos, Apr 16 2022 *)

Prog

(PARI) { S1 = [4, 2, 0, 1; 2, 8, 3, 1; 0, 3, 10, 5; 1, 1, 5, 10];
(dyn_th(N) = 1 + 2 * x * Ser(qfrep(S1, N, 1))); th = dyn_th(50);
a(n) = if(n >= #th - 2, th = dyn_th(2*n)); polcoeff(th, n); };
\\ Ben Mares, Apr 04 2022
(PARI) {a(n) = my(G = [4, 2, 0, 1; 2, 8, 3, 1; 0, 3, 10, 5; 1, 1, 5, 10]); if(n<0, 0, polcoeff(1 + 2*x*Ser(qfrep(G, n, 1)), n))}; /* Michael Somos, Apr 16 2022 */

Crossrefs

Sequence in context: A343468 A287846 A085623 * A369025 A190791 A002885

Adjacent sequences: A317962 A317963 A317964 * A317966 A317967 A317968

Keyword

nonn

Author

N. J. A. Sloane, Aug 26 2018

Extensions

More terms from Ben Mares, Mar 31 2022

Status

approved