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A374956 Decimal expansion of Muder's 1993 upper bound for the density of packing of unit spheres in the Euclidean 3-space.

%I A374956 #16 Jul 27 2024 03:58:28
%S A374956 7,7,3,0,5,5,8,9,6,5,7,6,9,0,8,8,9,0,5,5,0,2,1,7,5,5,7,0,1,5,2,9,0,4,
%T A374956 7,3,0,8,2,6,2,4,5,1,7,5,2,1,6,2,4,9,3,4,1,8,3,0,4,3,9,6,5,6,2,4,8,8,
%U A374956 9,2,7,5,9,6,8,6,5,0,8,8,8,0,5,0,9,1,0,5,2,5
%N A374956 Decimal expansion of Muder's 1993 upper bound for the density of packing of unit spheres in the Euclidean 3-space.
%C A374956 See A374772 for an improved bound.
%H A374956 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 A374956 Equals 4*Pi/(39*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 Corollary in Muder (1993), p. 352.
%F A374956 Equals (4/3)*Pi/A374955.
%e A374956 0.77305589657690889055021755701529047308262451752162...
%t A374956 Module[{beta, r, s},
%t A374956   s[p_] := Pi - 5*ArcTan[Sqrt[(1 - 2*r^2)/(p*r^2)]];
%t A374956   beta = 5*r*Sqrt[1 - 2*r^2]/(3*Sqrt[2]) + s[2]/6;
%t A374956   r = SolveValues[4/13*Pi == 2*s[3] - Sqrt[8/3]*s[2] && r > 0, r, Reals];
%t A374956   RealDigits[4*Pi/(39*beta), 10, 100][[1,1]]]
%Y A374956 Cf. A374772, A374837, A374955 (volume).
%K A374956 nonn,cons
%O A374956 0,1
%A A374956 _Paolo Xausa_, Jul 25 2024