A093766 Decimal expansion of Pi/(2*sqrt(3)).

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, 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, 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

Offset: 0

Author

Eric W. Weisstein, Apr 15 2004

Keywords

• nonn
• cons
• easy

Comments

Density of densest packing of equal circles in two dimensions (achieved for example by the A2 lattice).

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

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

Examples

			0.906899682117108925297039128821077866142033124046370287784942...
		

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer, 3rd. ed., 1998. See p. xix.
• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.7, p. 506.
• L. B. W. Jolley, Summation of Series, Dover (1961), Eq. (84) on page 16.
• Joel L. Schiff, The Laplace Transform: Theory and Applications, Springer-Verlag New York, Inc. (1999). See p. 149.
• David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 30.

Links

• J. H. Conway and N. J. A. Sloane, What are all the best sphere packings in low dimensions?, Discr. Comp. Geom., 13 (1995), 383-403.
• Xi Lin, Dirk Schmelter, Sadaf Imanian, and Horst Hintze-Bruening, Hierarchically Ordered alpha-Zirconium Phosphate Platelets in Aqueous Phase with Empty Liquid, Scientific Reports (2019) Vol. 9, Article No. 16389.
• R. J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015. See Table 22 for L(m=6,r=2,s=1).
• G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2.
• Michael I. Shamos, A catalog of the real numbers, (2007). See pp. 625-626.
• N. J. A. Sloane and Andrey Zabolotskiy, Table of maximal density of a packing of equal spheres in n-dimensional Euclidean space (some values are only conjectural).
• Eckard Specht, May 21 2012, The best known packings of equal circles in a circle (complete up to N=1500).
• László Fejes Tóth, An Inequality concerning polyhedra, Bull. Amer. Math. Soc. 54 (1948), 139-146. See p. 146.
• Eric Weisstein's World of Mathematics, Smoothed Octagon.
• Eric Weisstein's World of Mathematics, Circle Packing.
• Index entries for transcendental numbers.

Crossrefs

Related constants: A020769, A020789, A093825, A222066, A222067, A222068, A222069, A222070, A222071, A222072, A260646, (1/2)*A093602, A346585, A346584, A346583.

Cf. A072691, A381671.

Programs

Mathematica

RealDigits[Pi/(2 Sqrt[3]), 10, 111][[1]] (* Robert G. Wilson v, Nov 07 2012 *)

PARI

Pi/sqrt(12) \\ Charles R Greathouse IV, Oct 31 2014

Formula

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.

Equals Sum_{n>=1} 1/A134667(n). [Jolley]

Equals Sum_{n>=0} (-1)^n/A124647(n). [Jolley eq. 273]

Equals A000796 / A010469. - Omar E. Pol, Dec 09 2013

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

From Peter Bala, Feb 16 2015: (Start)

Equals 4*Sum_{n >= 0} 1/((6*n + 1)*(6*n + 5)).

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)

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

From Amiram Eldar, Aug 15 2020: (Start)

Equals Integral_{x=0..oo} 1/(x^2 + 3) dx.

Equals Integral_{x=0..oo} 1/(3*x^2 + 1) dx. (End)

Equals 1 + Sum_{k>=1} ( 1/(6*k+1) - 1/(6*k-1) ). - Sean A. Irvine, Jul 24 2021

For positive integer k, Pi/(2*sqrt(3)) = Sum_{n >= 0} (6*k + 4)/((6*n + 1)*(6*n + 6*k + 5)) - Sum_{n = 0..k-1} 1/(6*n + 5). - Peter Bala, Jul 10 2024

From Stefano Spezia, Jun 05 2025: (Start)

Equals Sum_{k>=0} (-1)^k/((k + 1)*(3*k + 1)).

Equals Integral_{x=0..oo} 1/(x^4 + x^2 + 1) dx.

Equals Integral_{x=0..oo} x^2/(x^4 + x^2 + 1) dx. (End)

Equals sqrt(A072691) = 3*A381671. - Hugo Pfoertner, Jun 05 2025

Extensions

Entry revised by N. J. A. Sloane, Feb 10 2013