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 A008350 Number of orbits of norm 2n vectors in E_8 lattice. 1
 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 (list; graph; refs; listen; history; text; internal format)
 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 EXTENSIONS Corrected by Noam D. Elkies, Apr 07 2008 STATUS approved

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Last modified February 5 10:11 EST 2023. Contains 360084 sequences. (Running on oeis4.)