A005134 Number of n-dimensional unimodular lattices (or quadratic forms). (Formerly M0219)

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

0,9

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

Table of n, a(n) for n=0..28.

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.

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

Crossrefs

Cf. A054907, A054908, A054909, A054911.

Sequence in context: A153683 A275882 A005139 * A081651 A269333 A029193

Adjacent sequences: A005131 A005132 A005133 * A005135 A005136 A005137

Keyword

nonn,nice,hard

Author

N. J. A. Sloane

Extensions

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

Status

approved