login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A267040 Decimal expansion of sqrt(8)*arctan(sqrt(2)/5). 3

%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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 29 04:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)