A054911 Number of n-dimensional odd unimodular lattices (or quadratic forms).

0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 6, 9, 13, 16, 28, 40, 68, 117, 273, 665, 2566, 17059, 374062

Offset

0,10

Comments

a(n) is also the class number of Z^n (the standard lattice with the identity as the basis), as every n-dimensional odd unimodular lattice lies in the same genus as Z^n. - Robin Visser, Jan 24 2025

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.

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, Cogent Seminar, Jul 05 2021.

Gaëtan Chenevier, Unimodular hunting, Modular Forms Workshop, Oberwolfach online, Feb 2021.

Gaëtan Chenevier, Unimodular Hunting, arXiv:2410.18788 [math.NT], 2024.

Steven R. Finch, Minkowski-Siegel mass constants [Broken link]

Steven R. Finch, Minkowski-Siegel mass constants

Oliver D. King, A mass formula for unimodular lattices with no roots, Math. Comp., 72 (2003), no. 242, 839-863. See Table 3 page 854.

Formula

If 8 divides n, then a(n) = A005134(n) - A054909(n/8), otherwise a(n) = A005134(n). - Robin Visser, Jan 24 2025

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

Prog

(Magma)
function a(n)
    if n lt 3 then return Min(1, n); end if;
    L := NumberFieldLattice(QNF(), n);
    return #GenusRepresentatives(L);
end function;  // Robin Visser, Jan 24 2025

Crossrefs

Cf. A005134, A054907, A054908, A054909, A241121, A241122.

Sequence in context: A018122 A074732 A089046 * A367215 A137267 A341451

Adjacent sequences: A054908 A054909 A054910 * A054912 A054913 A054914

Keyword

nonn,nice,hard,more,changed

Author

N. J. A. Sloane, May 23 2000

Status

approved