A005134 Number of n-dimensional unimodular lattices (or quadratic forms).

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 5, 8, 9, 13, 16, 28, 40, 68, 117, 297, 665, 2566, 17059, 374062

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• hard

Comments

King gives the lower bounds a(29) >= 37938009 and a(30) >= 20169641025. - Robin Visser, Feb 08 2025

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 49.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Bill Allombert and Gaëtan Chenevier, Unimodular Hunting II, arXiv:2410.19569 [math.NT], 2024.
• Gaëtan Chenevier, Unimodular Hunting, arXiv:2410.18788 [math.NT], 2024.
• Oliver D. King, A mass formula for unimodular lattices with no roots, Math. Comp., 72 (2003), no. 242, 839-863. See page 854.

Crossrefs

Cf. A054907, A054908, A054909, A054911.

Formula

If 8 divides n, then a(n) = A054911(n) + A054909(n/8), otherwise a(n) = A054911(n). - Robin Visser, Jan 24 2025

a(n) >= 2*A241121(n)/A241122(n). - Robin Visser, Feb 08 2025

Extensions

a(26)-a(28) added from Bill Allombert's and Gaëtan Chenevier's computations by Robin Visser, Jan 24 2025