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

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

Offset

1,4

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

Table of n, a(n) for n=1..8.

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).

Sequence in context: A145499 A221310 A082611 * A018324 A263803 A178175

Adjacent sequences: A033686 A033687 A033688 * A033690 A033691 A033692

Keyword

nonn,nice,hard,more

Author

N. J. A. Sloane

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).

Status

approved