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
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Aug 26 2018
EXTENSIONS
More terms from Ben Mares, Mar 31 2022
STATUS
approved