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%I
%S 7,8,6,2,2,2,4,1,8,2,6,2,6,6,8,9,8,2,1,4,2,4,9,8,3,8,4,1,3,2,5,9,8,8,
%T 8,1,0,7,1,8,2,8,2,9,3,7,1,7,0,8,9,5,1,7,5,3,0,8,3,2,1,4,1,7,1,8,6,0,
%U 9,9,4,5,7,5,8,8,1,2,6,3,2,6,4,8,5,3,6,7,4,7,7,9,6,7,2,2,0,7,9,3
%N Decimal expansion of 486^(1/3).
%C The cube root of 486 arises in Bezdek's proof on contact numbers for congruent sphere packings. Abstract: Let C(n) denote the largest number of touching pairs in a packing of n>1 congruent spheres in Euclidean 3-space. We prove that 0.695 < (6n-C(n))/(n^(2/3))< CubeRoot(486) = 7.862... for all n = (k(2k^2+1))/3 with k => 2.
%D K. Bezdek, On a stronger form of Rogers’ lemma and the minimum surface area of Voronoi cells in unit ball packings, J. reine angew. Math. 518 (2000), 131-143.
%D K. Bezdek, On the maximum number of touching pairs in a finite packing of translates of a convex body, J. Combin. Theory Ser. A 98/1 (2002), 192-200.
%D H. Harborth, Losung zu Problem 664A, Elem. Math. 29 (1974), 14-15.
%D G. A. Kabatiansky and V.I. Levenshtein, Bounds for packings on a sphere and in space, Problemy Peredachi Informatsii 14 (1978), 3-25.
%D G. Kuperberg and O. Schramm, Average kissing numbers for noncongruent sphere packings, Math. Res. Lett. 1/3 (1994), 339-344.
%D C. A. Rogers, Packing and Covering, Camb. Univ. Press, 1964.
%H Karoly Bezdek, <a href="http://arxiv.org/abs/1102.1198">Contact numbers for congruent sphere packings</a>, Feb 6, 2011.
%H Cf. <a href="http://oeis.org/search?q=sphere+packing">"Sphere packing"</a> in the OEIS
%e 7.86222418262668982142498384132598881...
%K nonn,easy,cons
%O 1,1
%A _Jonathan Vos Post_, Feb 08 2011
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