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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 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 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 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), 10, 111][] (* 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.)