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A093766 Decimal expansion of Pi/(2*sqrt(3)). 22
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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

REFERENCES

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer, 3rd. ed., 1998. See p. xix.

L. B. W. Jolley, Summation of Series, Dover (1961), Eq. (84) on page 16.

LINKS

Table of n, a(n) for n=0..101.

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, 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..., arXiv:1008.2547, 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

N. J. A. Sloane, Table of maximal density of a packing of equal spheres in n-dimensional Euclidean space (for n>3 the 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

FORMULA

Pi/(2*sqrt(3)) = (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. 271]

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

Continued fraction expansion: Pi/(2*sqrt(3)) = 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)

Pi/(2*sqrt(3)) = 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)

EXAMPLE

0.906899682117108925297039128821077866142033124046370287784942...

MATHEMATICA

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

PROG

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

CROSSREFS

Related constants: A020769, A020789, A093825, A222066, A222067, A222068, A222069, A222070, A222071, A222072, A222073, A222074, A222075, (1/2)*A093602.

Sequence in context: A225464 A296566 A198213 * A097674 A309823 A196549

Adjacent sequences:  A093763 A093764 A093765 * A093767 A093768 A093769

KEYWORD

nonn,cons,easy

AUTHOR

Eric W. Weisstein, Apr 15 2004

EXTENSIONS

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

STATUS

approved

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Last modified April 22 06:01 EDT 2021. Contains 343161 sequences. (Running on oeis4.)