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%I #31 Apr 17 2022 18:58:53
%S 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,
%T 14,32,10,46,8,46,30,28,28,62,34,32,40,38,28,48,28,60,50,48,32,50,28,
%U 62,34,52,26,68,30,62,56,68,38,110,28,50,64,86,60,72,50,56,34,88,50,138
%N Representation numbers (or theta series) for either one of Schiemann's pair of four-dimensional positive definite quadratic forms with the same representation numbers.
%C a(n) is the number of ways either form represents 2n.
%C 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
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, Third edition, p. xxix.
%H Ben Mares, <a href="/A317965/b317965.txt">Table of n, a(n) for n = 0..20000</a>
%H J. H. Conway and N. J. A. Sloane, <a href="http://NeilSloane.com/doc/nipp.pdf">Four-Dimensional Lattices With the Same Theta Series</a>, Duke Math. J., 66 (International Mathematics Research Notices Number 4) (1992), pp. 93-96. (Pages are in reverse order)
%H A. G. Earnest and G. Nipp, <a href="https://mr.math.ca/article/on-the-theta-series-of-positive-quaternary-quadratic-forms/">On the theta series of positive quaternary quadratic forms</a>, CR Math. Rep. Acad. Sci. Canada 13 (1991): 33-38. See p. 36.
%H Alexander Schiemann, <a href="https://doi.org/10.1007/BF01189584">Ein Beispiel positiv definiter quadratischer Formen der Dimension 4 mit gleichen Darstellungszahlen</a>, Archiv der Mathematik, 54.4 (1990): 372-375.
%e 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
%t 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 *)
%o (PARI) { S1 = [4, 2, 0, 1; 2, 8, 3, 1; 0, 3, 10, 5; 1, 1, 5, 10];
%o (dyn_th(N) = 1 + 2 * x * Ser(qfrep(S1, N, 1))); th = dyn_th(50);
%o a(n) = if(n >= #th - 2, th = dyn_th(2*n)); polcoeff(th, n);};
%o \\ _Ben Mares_, Apr 04 2022
%o (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 */
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Aug 26 2018
%E More terms from _Ben Mares_, Mar 31 2022
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