A394103 Decimal expansion of the mean distance between two points uniformly and independently selected at random in a regular octahedron of unit volume.

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

Offset

0,1

Links

Table of n, a(n) for n=0..105.

Dominik Beck, Mean distance in polyhedra, arXiv:2309.13177 [math.PR], 2023.

Dominik Beck, Mean distance in polyhedra, lecture, 2024.

Dominik Beck, Random polytopes, doctoral thesis, Mathematical Institute of Charles University, Prague, 2025.

Formula

Equals (3/4)^(1/3) * (4/105 + 13*sqrt(2)/105 - 4*Pi/45 + 109*log(3)/(630*sqrt(2)) + 16*arccot(sqrt(2))/315 + 158*arccoth(sqrt(2))*sqrt(2)/315).

Example

0.658530729255500441731077133117686001234565908435197...

Mathematica

RealDigits[(3/4)^(1/3)*(4/105 + (13*Sqrt[2])/105 - (4*Pi)/45 + (109*Log[3])/(630*Sqrt[2]) + (16*ArcCot[Sqrt[2]])/315 + (158*ArcCoth[Sqrt[2]]*Sqrt[2])/315), 10, 120][[1]]

Prog

(PARI) (3/4)^(1/3) * (4/105 + 13*sqrt(2)/105 - 4*Pi/45 + 109*log(3)/(630*sqrt(2)) + 16*atan(1/sqrt(2))/315 + 158*atanh(1/sqrt(2))*sqrt(2)/315)

Crossrefs

Analogous constants: A073012 (cube), A366019 (regular tetrahedron), this constant (regular octahedron), A394104 (regular icosahedron), A394105 (regular dodecahedron).

Sequence in context: A329247 A394535 A133618 * A194599 A253300 A339253

Adjacent sequences: A394100 A394101 A394102 * A394104 A394105 A394106

Keyword

nonn,cons

Author

Amiram Eldar, Mar 10 2026

Status

approved