cp's OEIS Frontend

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

%I A054911 #31 Feb 10 2025 04:38:01
%S A054911 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,
%T A054911 17059,374062
%N A054911 Number of n-dimensional odd unimodular lattices (or quadratic forms).
%C A054911 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
%C A054911 King gives the lower bounds a(29) >= 37938009 and a(30) >= 20169641025. - _Robin Visser_, Feb 08 2025
%D A054911 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 49.
%H A054911 Bill Allombert and Gaëtan Chenevier, <a href="https://arxiv.org/abs/2410.19569">Unimodular Hunting II</a>, arXiv:2410.19569 [math.NT], 2024.
%H A054911 Gaëtan Chenevier, <a href="https://drive.google.com/file/d/1d2mrQYuUdj3bLsmUP57J0LL3I1kRRS_v/view">Unimodular hunting</a>, Cogent Seminar, Jul 05 2021.
%H A054911 Gaëtan Chenevier, <a href="http://gaetan.chenevier.perso.math.cnrs.fr/Unimodular_hunting_oberwolfach.pdf">Unimodular hunting</a>, Modular Forms Workshop, Oberwolfach online, Feb 2021.
%H A054911 Gaëtan Chenevier, <a href="https://arxiv.org/abs/2410.18788">Unimodular Hunting</a>, arXiv:2410.18788 [math.NT], 2024.
%H A054911 Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/">Minkowski-Siegel mass constants</a> [Broken link]
%H A054911 Steven R. Finch, <a href="https://oeis.org/A241121/a241121.pdf">Minkowski-Siegel mass constants</a>
%H A054911 Oliver D. King, <a href="https://doi.org/10.1090/S0025-5718-02-01455-2">A mass formula for unimodular lattices with no roots</a>, Math. Comp., 72 (2003), no. 242, 839-863.  See Table 3 page 854.
%F A054911 If 8 divides n, then a(n) = A005134(n) - A054909(n/8), otherwise a(n) = A005134(n). - _Robin Visser_, Jan 24 2025
%F A054911 a(n) >= 2*A241121(n)/A241122(n). - _Robin Visser_, Feb 08 2025
%o A054911 (Magma)
%o A054911 function a(n)
%o A054911     if n lt 3 then return Min(1,n); end if;
%o A054911     L := NumberFieldLattice(QNF(), n);
%o A054911     return #GenusRepresentatives(L);
%o A054911 end function;  // _Robin Visser_, Jan 24 2025
%Y A054911 Cf. A005134, A054907, A054908, A054909, A241121, A241122.
%K A054911 nonn,nice,hard,more
%O A054911 0,10
%A A054911 _N. J. A. Sloane_, May 23 2000