%I #28 Sep 08 2022 08:46:15
%S 7,7,9,6,3,5,5,7,0,0,4,4,2,5,2,9,3,7,8,2,5,4,5,0,8,2,4,9,9,3,2,2,9,0,
%T 3,0,0,1,1,1,1,5,9,5,3,9,5,2,5,4,8,4,2,5,5,2,7,3,1,7,0,3,7,0,0,1,1,8,
%U 3,5,2,4,4,9,0,7,6,1,7,9,0,5,1,0,5,9,4
%N Decimal expansion of sqrt(8)*arctan(sqrt(2)/5).
%C This constant is the packing density of a regular tetrahedron. That is, let S be a regular tetrahedron of edge length 2 and let B the part of S that lies within distance 1 of some vertex. Then this constant is the ratio of the volume of B to the volume of S.
%H G. C. Greubel, <a href="/A267040/b267040.txt">Table of n, a(n) for n = 0..10000</a>
%H Thomas C. Hales, <a href="http://dx.doi.org/10.4007/annals.2005.162.1065">A proof of the Kepler conjecture</a>, Annals of Mathematics, Vol. 162, No. 3 (2005), pp. 1065-1185.
%H Carl A. Rogers, <a href="https://doi.org/10.1112/plms/s3-8.4.609">The packing of equal spheres</a>, Proceedings of the London Mathematical Society, Vol. 3, No. 4 (1958), pp. 609-620.
%F Equals sqrt(8)*arctan(sqrt(2)/5).
%F Equals 3*A093825 - 2*A266814.
%F Equals 3*sqrt(2)*(arccos(1/3) - Pi/3). - _Amiram Eldar_, Jun 26 2021
%e 0.77963557004425293782545082499322903001111595395254...
%t First@ RealDigits@ N[Sqrt[8] ArcTan[Sqrt[2]/5], 120] (* _Michael De Vlieger_, Jan 09 2016 *)
%o (PARI) sqrt(8)*atan(sqrt(2)/5) \\ _Michel Marcus_, Jan 09 2016
%o (Magma) SetDefaultRealField(RealField(100)); Sqrt(8)*Arctan(Sqrt(2)/5); // _G. C. Greubel_, Aug 19 2018
%Y Cf. A093825, A266814, A267033.
%K nonn,cons
%O 0,1
%A _Martin Renner_, Jan 09 2016
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