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A363822 a(n) is the conjectured number of stable distinct centroidal Voronoi tessellations (CVTs) of a unit disk with n generators (seeds).

%I A363822 #28 Dec 06 2023 14:27:26
%S A363822 1,1,1,1,2,2,2,2,3,3,3,3,4,5,6,6,5,5,5,6,9,10
%N A363822 a(n) is the conjectured number of stable distinct centroidal Voronoi tessellations (CVTs) of a unit disk with n generators (seeds).
%C A363822 Stable CVTs are local minimizers of the CVT function (see Hateley, Wei,and Chen article).
%C A363822 There are other CVTs which are saddle points.
%C A363822 Lloyd's process converges only to stable CVTs from which different with respect to rotation symmetry are selected.
%C A363822 An efficient two-step semi-manual algorithm is used to recognize identical patterns and a fast code for the Lloyd's process.
%C A363822 Code in Mathematica and details published on Github.
%D A363822 J. C. Hateley, H. Wei, and L. Chen, Fast Methods for Computing Centroidal Voronoi Tessellations, 2014 J Sci Comput DOI 10.1007/10915-014-9894-1
%D A363822 Yang Liu, Wenping Wang, Bruno Lévy, Feng Sun, Dong-Ming Yan, Lin Lu, and Chenglei Yang, On centroidal Voronoi tessellation—Energy smoothness and fast computation, ACM Transactions on Graphics, Volume 28, Issue 4, Article No. 101, pp. 1-17, 2009, DOI 10.1145/1559755.1559758
%D A363822 Lin Lu, F. Sun, and H. Pan, Global optimization Centroidal Voronoi Tessellation with Monte Carlo Approach, 2012 IEEECS Log Number TVCG-2011-03-0067.
%H A363822 Denis Ivanov, <a href="https://github.com/lesobrod/CVT">Github with code, explanations and results</a>.
%H A363822 Wikipedia, <a href="https://en.wikipedia.org/wiki/Centroidal_Voronoi_tessellation">Centroidal Voronoi tessellation </a> (unfortunately, article is a stub and contains inaccuracies).
%H A363822 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lloyd%27s_algorithm">Lloyd's algorithm</a>.
%e A363822 As initialization, clustering centers for a large number of points in the unit disk are used. For every set of centers, Lloyd's algorithm is iterated and all variants symmetric with respect to rotations are removed.
%Y A363822 Cf. A366544 (square).
%K A363822 nonn,hard,more
%O A363822 0,5
%A A363822 _Denis Ivanov_, Oct 18 2023