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Number of perfect quadratic forms or lattices in dimension n.
(Formerly M0862)
7

%I M0862 #44 May 04 2021 12:03:12

%S 1,1,1,2,3,7,33,10916

%N Number of perfect quadratic forms or lattices in dimension n.

%D 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).

%D J. Martinet, Les réseaux parfaits des espaces Euclidiens, Masson, Paris, 1996, p. 175.

%D J. Martinet, Perfect Lattices in Euclidean Spaces, Springer-Verlag, NY, 2003.

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

%D A. Schürmann, Enumerating perfect forms, Contemporary Math., 493 (2009), 359-377. [From _N. J. A. Sloane_, Jan 21 2010]

%H J. H. Conway and N. J. A. Sloane, <a href="http://www.jstor.org/stable/2398316">Low-dimensional lattices III: perfect forms</a>, Proc. Royal Soc. London, A 418 (1988), 43-80.

%H J. H. Conway and N. J. A. Sloane, <a href="http://neilsloane.com/doc/splag.html">Sphere Packings, Lattices and Groups</a>, Springer-Verlag, 3rd edition, 1999, see Preface to 3rd Ed., especially the page that was omitted by the publisher between pages xx and xxi!

%H Mathieu Dutour Sikiric, Achill Schürmann and Frank Vallentin, <a href="https://www.math.uni-magdeburg.de/lattice_geometry/perfect-forms-dim8.txt">Complete list of perfect forms in dimension 8</a>

%H M. Dutour Sikiric, A. Schürmann and F. Vallentin, <a href="http://arxiv.org/abs/math/0609388">Classification of eight-dimensional perfect forms</a>, arXiv:math/0609388 [math.NT], 2006-2009.

%H M. Dutour Sikiric, A. Schürmann and F. Vallentin, <a href="http://dx.doi.org/10.1090/S1079-6762-07-00171-0">Classification of eight-dimensional perfect forms</a>, Electron. Res. Announc. Amer. Math. Soc. 13 (2007), 21-32.

%H Mathieu Dutour Sikirić, Philippe Elbaz-Vincent, Alexander Kupers and Jacques Martinet, <a href="https://arxiv.org/abs/1910.11598">Voronoi complexes in higher dimensions, cohomology of GL_N(Z) for N >= 8 and the triviality of K8(Z)</a>, arXiv:1910.11598 [math.KT], 2019.

%H D.-O. Jaquet and F. Sigrist, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k56752189/f647.image">Formes quadratiques contiguës à D_7</a>, C. R. Acad. Sci. Paris Ser. I Math. 309 (1989), no. 10, 641-644.

%H G. Nebe, <a href="http://www.ams.org/journals/bull/2004-41-04/S0273-0979-04-01018-3/">Review of J. Martinet, Perfect Lattices in Euclidean Spaces</a>, Bull. Amer. Math. Soc., 41 (No. 4, 2004), 529-533.

%H Kristen Scheckelhoff, Kalani Thalagoda, and Dan Yasaki, <a href="https://arxiv.org/abs/2105.00593">Perfect Forms over Imaginary Quadratic Fields</a>, arXiv:2105.00593 [math.NT], 2021.

%H A. Schürmann, <a href="http://inst-mat.utalca.cl/qfc2007/Talks/schuermann.pdf">Enumerating perfect forms</a>, International Conference on the Algebraic and Arithmetic Theory of Quadratic Forms, 2007.

%H A. Schürmann, <a href="http://arxiv.org/abs/0901.1587">Enumerating perfect forms</a>, arXiv:0901.1587 [math.NT], 2009.

%H B. Venkov, <a href="https://jamartin.perso.math.cnrs.fr/Publications/venkovensmath.pdf">Réseaux et designs sphériques</a>, pp. 10-86 in Réseaux Euclidiens, Designs Sphériques et Formes Modulaires, ed. J. Martinet, L'Enseignement Mathématique, Geneva, 2001.

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

%K hard,more,nonn,nice

%O 1,4

%A _N. J. A. Sloane_

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