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A093766
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Decimal expansion of Pi/(2*sqrt(3)).
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17
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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
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OFFSET
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0,1
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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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.
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, 21 May 2012, The best known packings of equal circles in a circle (complete up to N=1500)
Eric Weisstein's World of Mathematics, Smoothed Octagon
Eric Weisstein's World of Mathematics, Circle Packing
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FORMULA
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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]
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)
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EXAMPLE
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0.906899682117108925297039128821077866142033124046370287784942...
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MATHEMATICA
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RealDigits[Pi/(2 Sqrt[3]), 10, 111][[1]] (* Robert G. Wilson v, Nov 07 2012 *)
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PROG
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(PARI) Pi/sqrt(12) \\ Charles R Greathouse IV, Oct 31 2014
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CROSSREFS
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Related constants: A020769, A020789, A093825, A222066, A222067, A222068, A222069, A222070, A222071, A222072, A222073, A222074, A222075, 1/2 * A093602.
Sequence in context: A131223 A225464 A198213 * A097674 A196549 A173164
Adjacent sequences: A093763 A093764 A093765 * A093767 A093768 A093769
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KEYWORD
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nonn,cons,easy
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AUTHOR
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Eric W. Weisstein, Apr 15 2004
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EXTENSIONS
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Entry revised by N. J. A. Sloane, Feb 10 2013
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STATUS
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approved
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