The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.



(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A004026 Number of perfect quadratic forms or lattices in dimension n.
(Formerly M0862)
1, 1, 1, 2, 3, 7, 33, 10916 (list; graph; refs; listen; history; text; internal format)



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.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

A. Schürmann, Enumerating perfect forms, Contemporary Math., 493 (2009), 359-377. [From N. J. A. Sloane, Jan 21 2010]


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

J. H. Conway and N. J. A. Sloane, Low-dimensional lattices III: perfect forms, Proc. Royal Soc. London, A 418 (1988), 43-80.

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!

Mathieu Dutour Sikiric, Achill Schürmann and Frank Vallentin, Complete list of perfect forms in dimension 8

M. Dutour Sikiric, A. Schürmann and F. Vallentin, Classification of eight-dimensional perfect forms, arXiv:math/0609388 [math.NT], 2006-2009.

M. Dutour Sikiric, A. Schürmann and F. Vallentin, Classification of eight-dimensional perfect forms, Electron. Res. Announc. Amer. Math. Soc. 13 (2007), 21-32.

Mathieu Dutour Sikirić, Philippe Elbaz-Vincent, Alexander Kupers and Jacques Martinet, Voronoi complexes in higher dimensions, cohomology of GL_N(Z) for N >= 8 and the triviality of K8(Z), arXiv:1910.11598 [math.KT], 2019.

D.-O. Jaquet and F. Sigrist, Formes quadratiques contiguës à D_7, C. R. Acad. Sci. Paris Ser. I Math. 309 (1989), no. 10, 641-644.

G. Nebe, Review of J. Martinet, Perfect Lattices in Euclidean Spaces, Bull. Amer. Math. Soc., 41 (No. 4, 2004), 529-533.

Kristen Scheckelhoff, Kalani Thalagoda, and Dan Yasaki, Perfect Forms over Imaginary Quadratic Fields, arXiv:2105.00593 [math.NT], 2021.

A. Schürmann, Enumerating perfect forms, International Conference on the Algebraic and Arithmetic Theory of Quadratic Forms, 2007.

A. Schürmann, Enumerating perfect forms, arXiv:0901.1587 [math.NT], 2009.

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.


Cf. A033689, A065535, A065536, A037075, A122079, A122080.

Sequence in context: A057677 A032148 A101484 * A135907 A027624 A165744

Adjacent sequences:  A004023 A004024 A004025 * A004027 A004028 A004029




N. J. A. Sloane


a(8) from the work of Mathieu Dutour Sikiric, Achill Schuermann and Frank Vallentin, Oct 05 2005



Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 7 15:02 EDT 2021. Contains 343650 sequences. (Running on oeis4.)