cp's OEIS Frontend

A374955 Decimal expansion of Muder's 1993 lower bound for the volume of any Voronoi polyhedron defined by a packing of unit spheres in the Euclidean 3-space.

%I A374955 #17 Jul 28 2024 00:14:56
%S A374955 5,4,1,8,4,8,2,9,6,2,6,6,0,7,2,3,2,9,4,1,4,4,5,7,2,5,2,0,9,3,2,4,6,4,
%T A374955 5,2,7,8,1,8,3,0,9,5,5,8,9,9,8,2,2,5,7,2,5,6,3,7,3,1,6,4,4,7,5,3,5,9,
%U A374955 9,8,3,8,9,9,2,1,6,9,9,6,0,3,8,8,7,9,8,6,2,8
%N A374955 Decimal expansion of Muder's 1993 lower bound for the volume of any Voronoi polyhedron defined by a packing of unit spheres in the Euclidean 3-space.
%C A374955 See A374753 (the dodecahedral conjecture) for an improved bound.
%H A374955 Douglas J. Muder, <a href="https://doi.org/10.1007/BF02573984">A New Bound on the Local Density of Sphere Packings</a>, Discrete & Computational Geometry, Vol. 10, 1993, pp. 351-375.
%F A374955 Equals 13*beta, where beta = 5*r*sqrt(1-2*r^2)/(3*sqrt(2)) + (1/6)*(Pi - 5*arctan(sqrt((1 - 2*r^2)/(2*r^2)))) and r is the positive solution to (4/13)*Pi = 2*(Pi - 5*arctan(sqrt((1 - 2*r^2)/(3*r^2)))) - sqrt(8/3)*(Pi - 5*arctan(sqrt((1 - 2*r^2)/(2*r^2)))). See Theorem in Muder (1993), p. 352.
%F A374955 Equals (4/3)*Pi/A374956.
%e A374955 5.4184829626607232941445725209324645278183095589982...
%t A374955 Module[{beta, r, s},
%t A374955   s[p_] := Pi - 5*ArcTan[Sqrt[(1 - 2*r^2)/(p*r^2)]];
%t A374955   beta = 5*r*Sqrt[1 - 2*r^2]/(3*Sqrt[2]) + s[2]/6;
%t A374955   r = SolveValues[4/13*Pi == 2*s[3] - Sqrt[8/3]*s[2] && r > 0, r, Reals];
%t A374955   RealDigits[13*beta, 10, 100][[1,1]]]
%Y A374955 Cf. A374771, A374753, A374956 (density).
%K A374955 nonn,cons
%O A374955 1,1
%A A374955 _Paolo Xausa_, Jul 25 2024