A008350 Number of orbits of norm 2n vectors in E_8 lattice.

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, 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, 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

Offset

0,5

Comments

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

References

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 123.

Links

N. J. A. Sloane, Table of n, a(n) for n = 0..1200 [Computed using Elkies's PARI/GP program]

J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).

Gabriele Nebe and N. J. A. Sloane, Home page for this lattice

Kaiwen Sun and Haowu Wang, Weyl invariant E8 Jacobi forms and E-strings, arXiv:2109.10578 [math.NT], 2021. See Table 5 p. 16.

Prog

(PARI)
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
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)
{ orbit_counts(N) =
c = vector(N);
v = vector(8, n, 0);
j = 1;
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]++); );
return(concat(1, c)) }
orbit_counts(100) \\ Noam D. Elkies, Apr 07 2008

Crossrefs

Sequence in context: A282091 A354110 A015718 * A019556 A165640 A082892

Adjacent sequences: A008347 A008348 A008349 * A008351 A008352 A008353

Keyword

nonn

Author

N. J. A. Sloane and J. H. Conway

Extensions

Corrected by Noam D. Elkies, Apr 07 2008

Status

approved