%I #27 Sep 23 2021 02:25:34
%S 1,1,1,1,2,1,1,2,2,2,2,2,2,3,2,2,4,3,3,4,3,3,4,4,3,5,4,5,6,5,3,6,6,5,
%T 6,6,6,8,6,6,7,7,6,10,8,7,8,9,7,10,9,9,11,11,8,11,10,10,12,12,9,13,11,
%U 13,14,13,10,17,14,12,13,15,13,17,15,15,17,18,13,19,16,16,18,21,15,20,18,19
%N Number of orbits of norm 2n vectors in E_8 lattice.
%C Since Aut(E8) is a reflection group one can compute this using nonnegative combinations of the basis dual to the simple roots, since these are the lattice vectors in a fundamental domain and so include a unique representative of each orbit. - _Noam D. Elkies_, Apr 07 2008
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 123.
%H N. J. A. Sloane, <a href="/A008350/b008350.txt">Table of n, a(n) for n = 0..1200</a> [Computed using Elkies's PARI/GP program]
%H J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (<a href="http://neilsloane.com/doc/Me220.pdf">pdf</a>).
%H Gabriele Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/E8.html">Home page for this lattice</a>
%H Kaiwen Sun and Haowu Wang, <a href="https://arxiv.org/abs/2109.10578">Weyl invariant E8 Jacobi forms and E-strings</a>, arXiv:2109.10578 [math.NT], 2021. See Table 5 p. 16.
%o (PARI)
%o M = 2*matid(8); for(i=1,6, M[i,i+1] = M[i+1,i] = -1); M[3,8] = M[8,3] = -1; \\ M is now the Gram matrix for the simple roots of E8
%o M = 1/M; \\ M is now the Gram matrix for the dual basis to the simple roots; their nonnegative combinations are a fundamental domain for W(E8)
%o { orbit_counts(N) =
%o c = vector(N);
%o v = vector(8,n,0);
%o j = 1;
%o while(j<9, j = 1; v[1]++; k = v*M*v~; while(k>2*N, v[j]=0; j++; if(j<9, v[j]++; k=v*M*v~, k=0)); if(k, c[k/2]++););
%o return(concat(1,c)) }
%o orbit_counts(100) \\ _Noam D. Elkies_, Apr 07 2008
%K nonn
%O 0,5
%A _N. J. A. Sloane_ and _J. H. Conway_
%E Corrected by _Noam D. Elkies_, Apr 07 2008
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