%I A081314
%S A081314 24,12,48,20,8,12,16,4,120,4,24,12,24,4,6,2,8,2,4,6,4,2,2,20
%N A081314 Order of symmetry groups of n points on 3-dimensional sphere with the 
               volume enclosed by their convex hull maximized.
%C A081314 If more than one configuration with maximal volume exists for a given 
               n, the one with the largest symmetry group is chosen. Berman and 
               Hanes give optimality proofs for n<=8. Higher terms are only conjectures. 
               An independent verification of the results by Hardin, Sloane and 
               Smith has been performed by Pfoertner in 1992 for n<28. An archive 
               of the results with improvements for n=23,24 added in 2003 is available 
               at link. A conjectured continuation of the sequence starting with 
               n=28 is: 12,6,2,6,120,2,4,4,2,20,4,12,24,12,20,4,8,2,2,2,4,1,24
%D A081314 Joel D. Berman and Kitt Hanes, Volumes of Polyhedra Inscribed in the 
               Unit Sphere in E3. Mathematische Annalen 188, 78-84 (1970)
%H A081314 R. H. Hardin, N. J. A. Sloane and W. D. Smith, <a href="http://www.research.att.com/
               ~njas/maxvolumes">Maximal Volume Spherical Codes</a>
%H A081314 Hugo Pfoertner, <a href="http://www.randomwalk.de/sphere/volmax">Maximal 
               Volume Arrangements of Points on Sphere.</a> Visualizations for n<=21.
%H A081314 Hugo Pfoertner, <a href="http://www.randomwalk.de/sphere/volmax/volmax.zip">
               Maximal Volume Arrangements: Archive</a>
%e A081314 a(12)=120 because the order of the point group of the icosahedron, which 
               is also the best known arrangement for the maximal volume problem 
               is 120. a(7)=20 because the double 7-pyramid proved optimal by Berman 
               and Hanes has dihedral symmetry order 20.
%Y A081314 Number of distinct edges in convex hull: A081366. Symmetry groups for 
               Tammes problem: A080865.
%Y A081314 Sequence in context: A079341 A080865 A040555 this_sequence A119872 A002550 
               A075605
%Y A081314 Adjacent sequences: A081311 A081312 A081313 this_sequence A081315 A081316 
               A081317
%K A081314 hard,nonn
%O A081314 4,1
%A A081314 Hugo Pfoertner (hugo(AT)pfoertner.org), Mar 19 2003