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%I #13 Oct 26 2018 02:47:33
%S 1,1,1,1,3,222
%N Number of different primitive polyhedral types of Voronoi regions of n-dimensional point lattices.
%C Or, number of combinatorial types of primitive n-dimensional parallelohedra.
%C Or, number of combinatorial types of Delaunay [Delone] decompositions of R^n.
%C Voronoi proved a(n) finite.
%D E. S. Barnes and N. J. A. Sloane, "The optimal lattice quantizer in three dimensions," SIAM J. Algebraic Discrete Methods vol. 4 (Mar. 1983) 30-41.
%D J. H. Conway, The Sensual Quadratic Form.
%D J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VI: Voronoi Reduction of Three-Dimensional Lattices, Proc. Royal Soc. London, Series A, 436 (1992), 55-68.
%D P. Engel, Investigations of parallelohedra in Rd, in: Voronoi's Impact on Modern Science, P. Engel and H. Syta (eds), Institute of Mathematics, Kyiv, 1998, Vol. 2, pp. 2260.
%D P. Engel, The contraction types of parallelohedra in E^5, Acta Crystallogr., A56 (2000), 491-496.
%D P. Engel and V. Grishukhin, There are exactly 222 L-types of primitive five-dimensional lattices. European J. Combin. 23 (2002), 275-279.
%D S. S. Ryshkov and E. P. Baranovskii, "C-types of n-dimensional lattices and 5-dimensional primitive parellohedra (with an application to the theory of coverings)" Proc. Steklov Inst. Math., 137 (1975) Trudy Mat. Inst. Steklov., 137 (1975)
%D M. I. Stogrin, Regular Dirichlet-Voronoi partitions for the second triclinic group, Trudy Matematicheskogo Instituta imeni V. A. Steklova, 123 (1973) = Proceedings of the Steklov Institute of Mathematics, 123 (1973).
%D G. F. Voronoi, "Studies of primitive parallelotopes", Collected Works, 2, Kiev (1952) pp. 239-368 (In Russian).
%H Mathieu Dutour Sikiric, Alexey Garber, <a href="https://arxiv.org/abs/1810.10911">Periodic triangulations of Z^n</a>, arXiv:1810.10911 [math.CO], 2018.
%H Russian Math. Encyclopedia, <a href="http://eom.springer.de/V/v096920.htm">Voronoi</a>
%H Achill Schuermann and Frank Vallentin, <a href="http://arxiv.org/abs/math/0403272">Computational Approaches to Lattice Packing and Covering Problems</a>, arXiv:math/0403272v3 [math.MG], 2004-2005.
%e a(2)=1 because the hexagon is the only allowed type (quadrilateral is a degenerate hexagon). a(3)=1 because the truncated octahedron is the only allowed type. - _Warren D. Smith_, Dec 27 2007
%Y Cf. A071880, A071882.
%K nonn,hard,nice
%O 0,5
%A _N. J. A. Sloane_, Jun 10 2002, Jul 03 2008