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Search: id:A008350
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| A008350 |
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Number of orbits of norm 2n vectors in E_8 lattice. |
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+0
1
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| 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; listen)
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OFFSET
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0,5
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COMMENT
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Comment from Noam Elkies, Apr 07 2008: 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.
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REFERENCES
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J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 123.
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LINKS
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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 (Abstract, pdf, ps).
G. Nebe and N. J. A. Sloane, Home page for this lattice
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PROGRAM
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(PARI/GP program from Noam Elkies, Apr 07 2008) 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)
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CROSSREFS
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Sequence in context: A062501 A067390 A015718 this_sequence A019556 A165640 A082892
Adjacent sequences: A008347 A008348 A008349 this_sequence A008351 A008352 A008353
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com) and J. H. Conway (conway(AT)math.princeton.edu)
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EXTENSIONS
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Corrected by Noam Elkies, Apr 07 2008
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