A093825 -id:A093825 - cp's OEIS frontend

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

Eric W. Weisstein, Apr 15 2004

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

			0.906899682117108925297039128821077866142033124046370287784942...

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.

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.

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

Cf. A072691, A381671.

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

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

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

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

A019673 Decimal expansion of Pi/6.

Original entry on oeis.org

5, 2, 3, 5, 9, 8, 7, 7, 5, 5, 9, 8, 2, 9, 8, 8, 7, 3, 0, 7, 7, 1, 0, 7, 2, 3, 0, 5, 4, 6, 5, 8, 3, 8, 1, 4, 0, 3, 2, 8, 6, 1, 5, 6, 6, 5, 6, 2, 5, 1, 7, 6, 3, 6, 8, 2, 9, 1, 5, 7, 4, 3, 2, 0, 5, 1, 3, 0, 2, 7, 3, 4, 3, 8, 1, 0, 3, 4, 8, 3, 3, 1, 0, 4, 6, 7, 2, 4, 7, 0, 8, 9, 0, 3, 5, 2, 8, 4, 4

Offset: 0

N. J. A. Sloane

From Omar E. Pol, Aug 30 2007: (Start)
Pi/6 = Volume of the inscribed ellipsoid / (Volume of the cuboid (If L1>L2>L3)).
Pi/6 = Volume of the inscribed spheroid / (Volume of the cuboid (If L1>(L2=L3))).
Pi/6 = Volume of the inscribed spheroid / (Volume of the cuboid (If L1<(L2=L3))).
Pi/6 = Volume of the inscribed sphere / (Volume of the regular hexahedron (Or cube)).  (End)
Pi/6 = Surface area of the inscribed sphere / (surface area of the regular hexahedron (or cube)). - Omar E. Pol, Nov 13 2007
Decimal expansion of arctan(sqrt(1/3)). - Clark Kimberling, Sep 23 2011
Also, decimal expansion of sum( k>=1, (-120+329*k+568*k^2)/(k*(1+k)*(1+2*k)*(1+4*k)*(3+4*k)*(5+4*k)) ). - Bruno Berselli, Dec 01 2013
Atomic packing factor (APF) of the simple cubic lattice filled with spheres of the same diameter (unique example among chemical elements: polonium crystal). - Stanislav Sykora, Sep 29 2014

			Pi/6 = 0.5235987755982988730771072305465838140328615665625176368291574...

Ian Stewart, Professor Stewart's Cabinet of Mathematical Curiosities, Basic Books, a member of the Perseus Books Group, NY, 2009, "A Constant Bore", pp. 49-50 & 264-266.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Atomic packing factor
Index entries for transcendental numbers

Cf. A000796, A132696.

Cf. APF's of other crystal lattices: A093825 (hcp,fcc), A247446 (diamond cubic).

C := ComplexField(); [Pi(C)/6]; // G. C. Greubel, Nov 18 2017

Mathematica

RealDigits[N[Pi/6,6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *) RealDigits[Pi/6,10,120][[1]] (* Harvey P. Dale, Oct 05 2024 *)

PARI

Pi/6 \\ Charles R Greathouse IV, Jul 07 2014

Formula

From Amiram Eldar, Aug 15 2020: (Start)

Equals Integral_{x=0..oo} 1/(x^2 + 9) dx.

Equals Integral_{x=0..oo} 1/(9*x^2 + 1) dx. (End)

Pi/6 = Sum_{n >= 1} i/(n*P(n,sqrt(-3))*P(n-1,sqrt(-3))), where i = sqrt(-1) and P(n,x) denotes the n-th Legendre polynomial. The first ten terms of the series gives the approximation Pi/6 = 0.52359877559(52...) correct to 11 decimal places - Peter Bala, Mar 16 2024

A222068 Decimal expansion of (1/16)*Pi^2.

Original entry on oeis.org

6, 1, 6, 8, 5, 0, 2, 7, 5, 0, 6, 8, 0, 8, 4, 9, 1, 3, 6, 7, 7, 1, 5, 5, 6, 8, 7, 4, 9, 2, 2, 5, 9, 4, 4, 5, 9, 5, 7, 1, 0, 6, 2, 1, 2, 9, 5, 2, 5, 4, 9, 4, 1, 4, 1, 5, 0, 8, 3, 4, 3, 3, 6, 0, 1, 3, 7, 5, 2, 8, 0, 1, 4, 0, 1, 2, 0, 0, 3, 2, 7, 6, 8, 7, 6, 1, 0, 8, 3, 7, 7, 3, 2, 4, 0, 9, 5, 1, 4, 4, 8, 9, 0, 0

Offset: 0

N. J. A. Sloane, Feb 10 2013

Conjectured to be density of densest packing of equal spheres in four dimensions (achieved for example by the D_4 lattice).
From Hugo Pfoertner, Aug 29 2018: (Start)
Also decimal expansion of Sum_{k>=0} (-1)^k*d(2*k+1)/(2*k+1), where d(n) is the number of divisors of n A000005(n).
Ramanujan's question 770 in the Journal of the Indian Mathematical Society (VIII, 120) asked "If d(n) denotes the number of divisors of n, show that d(1) - d(3)/3 + d(5)/5 - d(7)/7 + d(9)/9 - ... is a convergent series ...".
A summation of the first 2*10^9 terms performed by Hans Havermann yields 0.6168503077..., which is close to (Pi/4)^2=0.616850275...
(End)
From Robert Israel, Aug 31 2018: (Start)
Modulo questions about rearrangement of conditionally convergent series, which I expect a more careful treatment would handle, Sum_{k>=0} (-1)^k*d(2*k+1)/(2*k+1) should indeed be Pi^2/16.
Sum_{k>=0} (-1)^k d(2k+1)/(2k+1)
  = Sum_{k>=0} Sum_{2i+1 | 2k+1} (-1)^k/(2k+1)
(letting 2k+1=(2i+1)(2j+1): note that k == i+j (mod 2))
  = Sum_{i>=0} Sum_{j>=0} (-1)^(i+j)/((2i+1)(2j+1))
  = (Sum_{i>=0} (-1)^i/(2i+1))^2 = (Pi/4)^2. (End)
Volume bounded by the surface (x+y+z)^2-2(x^2+y^2+z^2)=4xyz, the ellipson (see Wildberger, p. 287). - Patrick D McLean, Dec 03 2020

			0.6168502750680849136771556874922594459571...

S. D. Chowla, Solution and Remarks on Question 770, J. Indian Math. Soc. 17 (1927-28), 166-171.
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. 507.
S. Ramanujan, Coll. Papers, Chelsea, 1962, Question 770, page 333.
G. N. Watson, Solution to Question 770, J. Indian Math. Soc. 18 (1929-30), 294-298.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
B. C. Berndt, Y. S. Choi and S. Y. Kang, The problems submitted by Ramanujan to the Journal of Indian Math. Soc., in: Continued fractions, Contemporary Math., 236 (1999), 15-56 (see Q770, JIMS VIII).
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.
Mathematics StackExchange, Sum_k (-1)^k tau(2k+1)/(2k+1).
G. Nebe and N. J. A. Sloane, Home page for D_4 lattice.
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).
N. J. Wildberger, Divine Proportions: Rational Trigonometry to Universal Geometry, Wild Egg Books, Sydney 2005.
Index entries for transcendental numbers.

Cf. A000005.

Related constants: A020769, A020789, A093766, A093825, A222066, A222067, A222069, A222070, A222071, A222072, A260646.

pi:=Pi(RealField(110)); Reverse(Intseq(Floor((1/16)*10^100*pi^2))); // Vincenzo Librandi, Feb 20 2017

RealDigits[N[Gamma[3/2]^4, 104]] (* Fred Daniel Kline, Feb 19 2017 *)
RealDigits[N[Pi^2/16, 100]][[1]] (* Vincenzo Librandi, Feb 20 2017 *)
Integrate[Boole[(x+y+z)^2-2(x^2+y^2+z^2)>4x y z],{x,0,1},{y,0,1},{z,0,1}] (* Patrick D McLean, Dec 03 2020 *)

(Pi/4)^2 \\ Charles R Greathouse IV, Oct 31 2014

Equals A003881^2. - Bruno Berselli, Feb 11 2013
Equals A123092+1/2. - R. J. Mathar, Feb 15 2013
Equals Integral_{x>0} x^2*log(x)/((1+x)^2*(1+x^2)) dx. - Jean-François Alcover, Apr 29 2013
Equals the Bessel moment integral_{x>0} x*I_0(x)*K_0(x)^3. - Jean-François Alcover, Jun 05 2016
Equals Sum_{k>=1} zeta(2*k)*k/4^k. - Amiram Eldar, May 29 2021

A020769 Decimal expansion of 1/sqrt(12) = 1/(2*sqrt(3)).

Original entry on oeis.org

2, 8, 8, 6, 7, 5, 1, 3, 4, 5, 9, 4, 8, 1, 2, 8, 8, 2, 2, 5, 4, 5, 7, 4, 3, 9, 0, 2, 5, 0, 9, 7, 8, 7, 2, 7, 8, 2, 3, 8, 0, 0, 8, 7, 5, 6, 3, 5, 0, 6, 3, 4, 3, 8, 0, 0, 9, 3, 0, 1, 1, 6, 3, 2, 4, 1, 9, 8, 8, 8, 3, 6, 1, 5, 1, 4, 6, 6, 6, 7, 2, 8, 4, 6, 8, 5, 7, 6, 9, 7, 7, 9, 2, 8, 7, 4, 7, 6, 2

Offset: 0

N. J. A. Sloane

Center density of densest packing of equal circles in two dimensions (achieved for example by the A2 lattice).
Let a equal the length of one side of an equilateral triangle and let b equal the radius of the circle inscribed in that triangle. This sequence gives the decimal expansion of b/a. - Christopher M. Tomaszewski (cmt1288(AT)comcast.net), Feb 20 2004
The constant (3+sqrt 3)/6, which is 0.5 larger than this, plays a role in Borsuk's conjecture. - Arkadiusz Wesolowski, Mar 17 2014

			0.28867513459481288225457439025097872782380087563506343800930116324198883615...

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

Ivan Panchenko, Table of n, a(n) for n = 0..1000
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 and Andrey Zabolotskiy, Table of maximal density of a packing of equal spheres in n-dimensional Euclidean space (some values are only conjectural).
Wikipedia, Borsuk's conjecture
Index entries for algebraic numbers, degree 2

Related constants: A020789, A093766, A093825, A222066, A222067, A222068, A222069, A222070, A222071, A222072, A260646.

RealDigits[N[1/Sqrt[12],200]] (* Vladimir Joseph Stephan Orlovsky, May 30 2010 *)

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

A260646 Decimal expansion of Pi^12/12!, the absolute density of the Leech lattice.

Original entry on oeis.org

0, 0, 1, 9, 2, 9, 5, 7, 4, 3, 0, 9, 4, 0, 3, 9, 2, 3, 0, 4, 7, 9, 0, 3, 3, 4, 5, 5, 6, 3, 6, 8, 5, 9, 5, 7, 6, 4, 0, 1, 6, 8, 4, 7, 1, 8, 1, 5, 0, 0, 0, 3, 0, 3, 3, 5, 2, 2, 3, 4, 6, 4, 7, 6, 1, 7, 3, 3, 1, 4, 9, 5, 6, 3, 4, 2, 5, 0, 9, 8, 5, 5, 3, 1, 4, 8, 7

Offset: 0

Felix Fröhlich, Nov 12 2015

			0.001929574309403923047903345563685957640168471815...

Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko and Maryna Viazovska, The sphere packing problem in dimension 24, Annals of Mathematics, 185 (2017), 1017-1033; arXiv:1603.06518 [math.NT], 2016-2017.
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, 1998 (see also Corrections and Updates), Chapter 4, Section 11.
G. Nebe and N. J. A. Sloane, Home page for Leech lattice.
A. Roberts, Properties of the Leech Lattice, 2006.
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).
Wikipedia, Leech lattice
Index entries for transcendental numbers

Densities of other lattices: A093766, A093825, A222068, A222069, A222070, A222071, A222072.

Related to Leech lattice: A008408, A323282.

RealDigits[N[Pi^12/12!, 120]]//First (* Michael De Vlieger, Nov 12 2015 *)

{ default(realprecision, 50080); x=Pi^12/12!; for(n=1, 100, d=floor(x); x=(x-d)*10; print1(d, ", ")) }

A222066 Decimal expansion of 1/sqrt(128).

Original entry on oeis.org

0, 8, 8, 3, 8, 8, 3, 4, 7, 6, 4, 8, 3, 1, 8, 4, 4, 0, 5, 5, 0, 1, 0, 5, 5, 4, 5, 2, 6, 3, 1, 0, 6, 1, 2, 9, 9, 1, 0, 6, 0, 4, 4, 9, 2, 2, 1, 1, 0, 5, 9, 2, 5, 4, 5, 7, 3, 5, 4, 2, 4, 8, 3, 6, 2, 4, 4, 2, 0, 7, 7, 9, 9, 0, 3, 8, 8, 1, 6, 8, 9, 9, 2, 8, 1, 4, 9, 2, 2, 0, 8, 9, 5, 4, 7, 7, 5, 9, 8, 2, 9, 5, 9, 3, 8

Offset: 0

N. J. A. Sloane, Feb 10 2013

Conjectured to be center density of densest packing of equal spheres in five dimensions (achieved for example by the D_5 lattice).

			.088388347648318440550105545263106129910604492211...

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

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 D_5 lattice
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).
Index entries for algebraic numbers, degree 2

Related constants: A020769, A020789, A093766, A093825, A222067, A222068, A222069, A222070, A222071, A222072, A260646.

Join[{0},RealDigits[1/Sqrt[128],10,120][[1]]] (* Harvey P. Dale, Sep 20 2023 *)

1/sqrt(128) \\ Charles R Greathouse IV, Oct 31 2014

Equals A020789/2. - R. J. Mathar, Jan 27 2021

A222070 Decimal expansion of (1/144)3^(1/2)Pi^3.

Original entry on oeis.org

3, 7, 2, 9, 4, 7, 5, 4, 5, 5, 8, 2, 0, 6, 4, 9, 3, 9, 5, 6, 3, 4, 7, 7, 5, 5, 8, 6, 7, 9, 9, 5, 8, 1, 0, 6, 3, 9, 3, 6, 6, 4, 7, 9, 7, 2, 6, 8, 3, 8, 7, 3, 6, 3, 1, 1, 1, 4, 0, 4, 0, 6, 5, 5, 9, 7, 2, 8, 3, 1, 7, 2, 0, 2, 9, 6, 8, 3, 2, 1, 9, 5, 2, 2, 5, 2, 6, 7, 2, 1, 6, 3, 5, 3, 4, 0, 5, 4, 2, 7, 6

Offset: 0

N. J. A. Sloane, Feb 10 2013

Conjectured to be density of densest packing of equal spheres in six dimensions (achieved for example by the E_6 lattice).

			0.3729475455820649395634775586799581063936647972683873631...

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. 507.

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 E_6 lattice.
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).
Index entries for transcendental numbers.

Related constants: A020769, A020789, A093766, A093825, A222066, A222067, A222068, A222069, A222071, A222072, A260646.

RealDigits[Sqrt[3]*Pi^3/144, 10, 120][[1]] (* Amiram Eldar, Jun 28 2023 *)

Pi^3*sqrt(3)/144 \\ Charles R Greathouse IV, Oct 31 2014

A222071 Decimal expansion of (1/105)*Pi^3.

Original entry on oeis.org

2, 9, 5, 2, 9, 7, 8, 7, 3, 1, 4, 5, 7, 1, 2, 5, 7, 3, 0, 9, 9, 7, 7, 4, 4, 2, 9, 2, 1, 0, 4, 8, 9, 4, 7, 8, 1, 1, 6, 4, 3, 1, 3, 1, 9, 6, 7, 5, 0, 9, 6, 2, 6, 3, 7, 5, 3, 7, 5, 7, 5, 0, 5, 7, 5, 0, 5, 3, 7, 0, 9, 4, 4, 5, 2, 0, 5, 4, 3, 4, 3, 2, 1, 4, 9, 2, 0, 9, 6, 2, 2, 1, 5, 2, 6, 5, 5, 8

Offset: 0

N. J. A. Sloane, Feb 10 2013

Conjectured to be density of densest packing of equal spheres in 7 dimensions (achieved for example by the E_7 lattice).

			0.295297873145712573099774429210489478116431319675...

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. 507.

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 E_7 lattice.
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).
Index entries for transcendental numbers.

Related constants: A020769, A020789, A093766, A093825, A222066, A222067, A222068, A222069, A222070, A222072, A260646.

RealDigits[Pi^3/105, 10, 120][[1]] (* Amiram Eldar, Jun 28 2023 *)

Pi^3/105 \\ Charles R Greathouse IV, Oct 31 2014

A222072 Decimal expansion of (1/384)*Pi^4.

Original entry on oeis.org

2, 5, 3, 6, 6, 9, 5, 0, 7, 9, 0, 1, 0, 4, 8, 0, 1, 3, 6, 3, 6, 5, 6, 3, 3, 6, 6, 3, 7, 6, 8, 3, 6, 2, 2, 7, 2, 1, 2, 8, 3, 2, 2, 5, 4, 3, 5, 5, 9, 5, 1, 6, 1, 8, 9, 8, 8, 1, 9, 7, 5, 5, 0, 4, 9, 4, 7, 1, 5, 7, 6, 9, 4, 1, 8, 8, 2, 0, 8, 2, 3, 4, 1, 1, 7, 7, 5, 6, 9, 5, 9, 2, 3, 8, 3, 5, 9, 1, 8, 1, 0, 1

Offset: 0

N. J. A. Sloane, Feb 10 2013

Conjectured to be density of densest packing of equal spheres in 8 dimensions (achieved for example by the E_8 lattice).

The above conjecture is true (cf. Viazovska, 2017). - Felix Fröhlich, Jan 08 2018

			.25366950790104801363656336637683622721283225435595161898819...

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. 507.

J. H. Conway and N. J. A. Sloane, What are all the best sphere packings in low dimensions?, Discrete & Computational Geometry, Vol. 13, No. 3-4 (1995), 383-403.
G. Nebe and N. J. A. Sloane, Home page for E_8 lattice.
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).
Maryna S. Viazovska, The sphere packing problem in dimension 8, Annals of Mathematics, Vol. 185, No. 3 (2017), 991-1015.
Maryna S. Viazovska, The sphere packing problem in dimension 8, arXiv:1603.04246 [math.NT], 2017.
Index entries for transcendental numbers.

Related constants: A020769, A020789, A093766, A093825, A222066, A222067, A222068, A222069, A222070, A222071, A260646.

RealDigits[Pi^4/ 384,10,120][[1]] (* Harvey P. Dale, Aug 11 2015 *)

Pi^4/384 \\ Charles R Greathouse IV, Oct 31 2014

Equals Sum_{n>=1} Sum_{k>=n} 1/(2*n - 1)^2/(2*k + 1)^2. - Geoffrey Critzer, Nov 03 2013

A222069 Decimal expansion of (1/30)2^(1/2)Pi^2.

Original entry on oeis.org

4, 6, 5, 2, 5, 7, 6, 1, 3, 3, 0, 9, 2, 5, 8, 6, 3, 5, 6, 1, 0, 5, 0, 4, 0, 6, 2, 4, 1, 1, 2, 9, 3, 6, 8, 5, 9, 9, 4, 6, 5, 7, 7, 5, 1, 3, 9, 6, 5, 3, 6, 1, 5, 7, 7, 4, 3, 5, 6, 6, 4, 4, 4, 5, 0, 1, 3, 2, 7, 1, 8, 4, 1, 8, 8, 8, 7, 1, 8, 1, 4, 3, 1, 1, 1, 6, 0, 0, 8, 9, 1, 5, 4, 0, 5, 4

Offset: 0

N. J. A. Sloane, Feb 10 2013

Conjectured to be density of densest packing of equal spheres in five dimensions (achieved for example by the D_5 lattice).

			.46525761330925863561050406241129368599465775139653615774...

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. 507.

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 D_5 lattice.
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).
Index entries for transcendental numbers.

Related constants: A020769, A020789, A093766, A093825, A222066, A222067, A222068, A222070, A222071, A222072, A260646.

RealDigits[(Sqrt[2] Pi^2)/30,10,120][[1]] (* Harvey P. Dale, Nov 07 2021 *)

Pi^2/sqrt(450) \\ Charles R Greathouse IV, Oct 31 2014

cp's OEIS Frontend

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

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A019673 Decimal expansion of Pi/6.

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A222068 Decimal expansion of (1/16)*Pi^2.

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A020769 Decimal expansion of 1/sqrt(12) = 1/(2*sqrt(3)).

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A260646 Decimal expansion of Pi^12/12!, the absolute density of the Leech lattice.

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A222066 Decimal expansion of 1/sqrt(128).

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A222070 Decimal expansion of (1/144)*3^(1/2)*Pi^3.

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A222070 Decimal expansion of (1/144)3^(1/2)Pi^3.

A222069 Decimal expansion of (1/30)2^(1/2)Pi^2.