A054909 Number of 8n-dimensional even unimodular lattices (or quadratic forms).

1, 1, 2, 24

Offset: 0

Author

N. J. A. Sloane, May 23 2000

Keywords

• nonn
• nice
• hard

Comments

King shows that a(4) >= 1162109024. - Charles R Greathouse IV, Nov 05 2013

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 49.

Links

• Steven R. Finch, Minkowski-Siegel mass constants [Broken link]
• Steven R. Finch, Minkowski-Siegel mass constants
• Oliver King, A mass formula for unimodular lattices with no roots, arXiv:math/0012231 [math.NT], 2000-2001; Mathematics of Computation 72:242 (2003), pp. 839-863.

Crossrefs

Cf. A005134, A054907, A054908, A054911.

Formula

a(n) = A005134(8*n) - A054911(8*n). - Robin Visser, Jan 24 2025

Extensions

The classical mass formula shows that the next term is at least 8*10^7.

Oliver King and Richard Borcherds (reb(AT)math.berkeley.edu) have recently improved this estimate and have shown that a(4), the number in dimension 32, is at least 10^9 (Jul 22 2000)