A033689 Number of extreme quadratic forms or lattices in dimension n.

1, 1, 1, 2, 3, 6, 30, 2408

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• hard
• more

Comments

A lattice is extreme if and only if it is perfect and eutactic. - Andrey Zabolotskiy, Feb 20 2021

References

• J. H. Conway and N. J. A. Sloane, Low-dimensional lattices III: perfect forms, Proc. Royal Soc. London, A 418 (1988), 43-80.
• M. Dutour Sikiric, A. Schuermann and F. Vallentin, Classification of eight-dimensional perfect forms, Preprint, 2006.
• P. M. Gruber, Convex and Discrete Geometry, Springer, 2007; p. 439
• D.-O. Jaquet, Classification des réseaux dans R^7 (via la notion de formes parfaites), Journées Arithmétiques, 1989 (Luminy, 1989). Asterisque No. 198-200 (1991), 7-8, 177-185 (1992).
• J. Martinet, Les réseaux parfaits des espaces Euclidiens, Masson, Paris, 1996, p. 175.
• J. Martinet, Perfect Lattices in Euclidean Spaces, Springer-Verlag, NY, 2003.
• G. Nebe, Review of J. Martinet, Perfect Lattices in Euclidean Spaces, Bull. Amer. Math. Soc., 41 (No. 4, 2004), 529-533.
• A. Schuermann, Enumerating perfect forms, Contemporary Math., 493 (2009), 359-377. [From N. J. A. Sloane, Jan 21 2010]

Links

• J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, 3rd edition, 1999, see Preface to 3rd Ed., especially the page that was omitted by the publisher between pages xx and xxi!
• D.-O. Jaquet and F. Sigrist, Formes quadratiques contigües à D_7, C. R. Acad. Sci. Paris Ser. I Math. 309 (1989), no. 10, 641-644.
• J. Martinet and B. Venkov, Les réseaux fortement eutactiques, pp. 112-132 in Réseaux Euclidiens, Designs Sphériques et Formes Modulaires, ed. J. Martinet, L'Enseignement Mathématique, Geneva, 2001.
• C. Riener, On extreme forms in dimension 8, J. Théor. Nombres Bordeaux 18 (2006), no. 3, 677-682.
• B. Venkov, Réseaux et designs sphériques, pp. 10-86 in Réseaux Euclidiens, Designs Sphériques et Formes Modulaires, ed. J. Martinet, L'Enseignement Mathématique, Geneva, 2001.

Crossrefs

Cf. A004026 (perfect), A037075 (eutactic).

Extensions

a(8) = 2408 was calculated by G. Nebe's student Cordian Riener - communicated by G. Nebe, Oct 11 2005. He found this number by checking the complete list of 10916 perfect lattices in 8 dimensions (see A004026).