A374956 Decimal expansion of Muder's 1993 upper bound for the density of packing of unit spheres in the Euclidean 3-space.

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

Offset

0,1

Comments

See A374772 for an improved bound.

Links

Table of n, a(n) for n=0..89.

Douglas J. Muder, A New Bound on the Local Density of Sphere Packings, Discrete & Computational Geometry, Vol. 10, 1993, pp. 351-375.

Formula

Equals 4*Pi/(39*beta), where beta = 5*r*sqrt(1-2*r^2)/(3*sqrt(2)) + (1/6)*(Pi - 5*arctan(sqrt((1 - 2*r^2)/(2*r^2)))) and r is the positive solution to (4/13)*Pi = 2*(Pi - 5*arctan(sqrt((1 - 2*r^2)/(3*r^2)))) - sqrt(8/3)*(Pi - 5*arctan(sqrt((1 - 2*r^2)/(2*r^2)))). See Corollary in Muder (1993), p. 352.

Equals (4/3)*Pi/A374955.

Example

0.77305589657690889055021755701529047308262451752162...

Mathematica

Module[{beta, r, s},
  s[p_] := Pi - 5*ArcTan[Sqrt[(1 - 2*r^2)/(p*r^2)]];
  beta = 5*r*Sqrt[1 - 2*r^2]/(3*Sqrt[2]) + s[2]/6;
  r = SolveValues[4/13*Pi == 2*s[3] - Sqrt[8/3]*s[2] && r > 0, r, Reals];
  RealDigits[4*Pi/(39*beta), 10, 100][[1, 1]]]

Crossrefs

Cf. A374772, A374837, A374955 (volume).

Sequence in context: A318302 A266271 A021568 * A199613 A136141 A264806

Adjacent sequences: A374953 A374954 A374955 * A374957 A374958 A374959

Keyword

nonn,cons

Author

Paolo Xausa, Jul 25 2024

Status

approved