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

Author

N. J. A. Sloane, May 23 2000

Keywords

• nonn
• nice
• hard
• more

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

• 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.

Crossrefs

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

Programs

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

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