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%I #84 Dec 30 2023 09:53:11
%S 9,0,6,8,9,9,6,8,2,1,1,7,1,0,8,9,2,5,2,9,7,0,3,9,1,2,8,8,2,1,0,7,7,8,
%T 6,6,1,4,2,0,3,3,1,2,4,0,4,6,3,7,0,2,8,7,7,8,4,9,4,2,4,6,7,6,9,4,0,6,
%U 1,5,9,0,5,6,3,1,7,6,9,4,1,8,4,2,0,6,2,4,9,4,1,0,6,0,3,0,0,8,4,4,2,8
%N Decimal expansion of Pi/(2*sqrt(3)).
%C Density of densest packing of equal circles in two dimensions (achieved for example by the A2 lattice).
%C The number gives the areal coverage (90.68... percent) of the close hexagonal (densest) packing of circles in the plane. The hexagonal unit cell is a rhombus of side length 1 and height sqrt(3)/2; the area of the unit cell is sqrt(3)/2 and the four parts of circles add to an area of one circle of radius 1/2, which is Pi/4. - _R. J. Mathar_, Nov 22 2011
%C Ratio of surface area of a sphere to the regular octahedron whose edge equals the diameter of the sphere. - _Omar E. Pol_, Dec 09 2013
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer, 3rd. ed., 1998. See p. xix.
%D L. B. W. Jolley, Summation of Series, Dover (1961), Eq. (84) on page 16.
%H J. H. Conway and N. J. A. Sloane, <a href="https://doi.org/10.1007/BF02574051">What are all the best sphere packings in low dimensions?</a>, Discr. Comp. Geom., 13 (1995), 383-403.
%H Xi Lin, Dirk Schmelter, Sadaf Imanian and Horst Hintze-Bruening, <a href="https://doi.org/10.1038/s41598-019-51934-y">Hierarchically Ordered alpha-Zirconium Phosphate Platelets in Aqueous Phase with Empty Liquid</a>, Scientific Reports (2019) Vol. 9, Article No. 16389.
%H R. J. Mathar, <a href="https://arxiv.org/abs/1008.2547">Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli</a>, arXiv:1008.2547, Table 22 for L(m=6,r=2,s=1).
%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A2.html">Home page for hexagonal (or triangular) lattice A2</a>
%H N. J. A. Sloane and Andrey Zabolotskiy, <a href="/A093825/a093825_1.txt">Table of maximal density of a packing of equal spheres in n-dimensional Euclidean space (some values are only conjectural)</a>.
%H Eckard Specht, May 21 2012, <a href="http://hydra.nat.uni-magdeburg.de/packing/cci/cci.html#Overview"> The best known packings of equal circles in a circle (complete up to N=1500)</a>
%H László Fejes Tóth, <a href="https://doi.org/10.1090/S0002-9904-1948-08969-8">An Inequality concerning polyhedra</a>, Bull. Amer. Math. Soc. 54 (1948), 139-146. See p. 146.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SmoothedOctagon.html">Smoothed Octagon</a>
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CirclePacking.html">Circle Packing</a>
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F Equals (5/6)*(7/6)*(11/12)*(13/12)*(17/18)*(19/18)*(23/24)*(29/30)*(31/30)*..., where the numerators are primes > 3 and the denominators are the nearest multiples of 6.
%F Equals Sum_{n>=1} 1/A134667(n). [Jolley]
%F Equals Sum_{n>=0} (-1)^n/A124647(n). [Jolley eq. 273]
%F Equals A000796 / A010469. - _Omar E. Pol_, Dec 09 2013
%F Continued fraction expansion: 1 - 2/(18 + 12*3^2/(24 + 12*5^2/(32 + ... + 12*(2*n - 1)^2/((8*n + 8) + ... )))). See A254381 for a sketch proof. - _Peter Bala_, Feb 04 2015
%F From _Peter Bala_, Feb 16 2015: (Start)
%F Equals 4*Sum_{n >= 0} 1/((6*n + 1)*(6*n + 5)).
%F Continued fraction: 1/(1 + 1^2/(4 + 5^2/(2 + 7^2/(4 + 11^2/(2 + ... + (6*n + 1)^2/(4 + (6*n + 5)^2/(2 + ... ))))))). (End)
%F The inverse is (2*sqrt(3))/Pi = Product_{n >= 1} 1 + (1 - 1/(4*n))/(4*n*(9*n^2 - 9*n + 2)) = (35/32) * (1287/1280) * (8075/8064) * (5635/5632) * (72819/72800) * ... = 1.102657790843585... - _Dimitris Valianatos_, Aug 31 2019
%F From _Amiram Eldar_, Aug 15 2020: (Start)
%F Equals Integral_{x=0..oo} 1/(x^2 + 3) dx.
%F Equals Integral_{x=0..oo} 1/(3*x^2 + 1) dx. (End)
%F Equals 1 + Sum_{k>=1} ( 1/(6*k+1) - 1/(6*k-1) ). - _Sean A. Irvine_, Jul 24 2021
%e 0.906899682117108925297039128821077866142033124046370287784942...
%t RealDigits[Pi/(2 Sqrt[3]), 10, 111][[1]] (* _Robert G. Wilson v_, Nov 07 2012 *)
%o (PARI) Pi/sqrt(12) \\ _Charles R Greathouse IV_, Oct 31 2014
%Y Related constants: A020769, A020789, A093825, A222066, A222067, A222068, A222069, A222070, A222071, A222072, A260646, (1/2)*A093602, A346585, A346584, A346583.
%K nonn,cons,easy
%O 0,1
%A _Eric W. Weisstein_, Apr 15 2004
%E Entry revised by _N. J. A. Sloane_, Feb 10 2013
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