A394104 Decimal expansion of the mean distance between two points uniformly and independently selected at random in a regular icosahedron of unit volume.

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

Offset

0,1

Links

Table of n, a(n) for n=0..104.

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 (1/2) * (9/5 - 3/sqrt(5))^(1/3) * (197/525 + 239/(525*sqrt(5)) - 44/525*sqrt(2 + 2/sqrt(5)) - (17226 + 6269*sqrt(5))*Pi/157500 - (2186 + 1413*sqrt(5))*arccot(phi)/15750 + (82 - 75*sqrt(5))*arccot(phi^2)/5250 + 4*(2139 + 881*sqrt(5))*arccosech(phi)/7875 + (15969 + 7151*sqrt(5))*arccoth(phi)/12600 + (4449 - 1685*sqrt(5))*log(3)/42000 - (75783 + 37789*sqrt(5))*log(5)/252000), where phi is the golden ratio (A001622).

Example

0.641312485510735885104521947992662588466399096164751...

Mathematica

With[{phi = GoldenRatio}, RealDigits[1/2*(9/5 - 3/Sqrt[5])^(1/3)*(197/525 + 239/(525*Sqrt[5]) - 44/525*Sqrt[2 + 2/Sqrt[5]] - ((17226 + 6269*Sqrt[5])*Pi)/157500 - ((2186 + 1413*Sqrt[5])*ArcCot[phi])/15750 + ((82 - 75*Sqrt[5])*ArcCot[phi^2])/5250 + (4*(2139 + 881*Sqrt[5])*ArcCsch[phi])/7875 + ((15969 + 7151*Sqrt[5])*ArcCoth[phi])/12600 + ((4449 - 1685*Sqrt[5])*Log[3])/42000 - ((75783 + 37789*Sqrt[5])*Log[5])/252000), 10, 120][[1]]]

Prog

(PARI) {my(phi = quadgen(5)); 1/2 * (9/5 - 3/sqrt(5))^(1/3) * (197/525 + 239/(525*sqrt(5)) - 44/525*sqrt(2 + 2/sqrt(5)) - (17226 + 6269*sqrt(5))*Pi/157500 - (2186 + 1413*sqrt(5))*atan(1/phi)/15750 + (82 - 75*sqrt(5))*atan(1/phi^2)/5250 + 4*(2139 + 881*sqrt(5))*asinh(1/phi)/7875 + (15969 + 7151*sqrt(5))*atanh(1/phi)/12600 + (4449 - 1685*sqrt(5))*log(3)/42000 - (75783 + 37789*sqrt(5))*log(5)/252000)}

Crossrefs

Cf. A001622.

Analogous constants: A073012 (cube), A366019 (regular tetrahedron), A394103 (regular octahedron), this constant (regular icosahedron), A394105 (regular dodecahedron).

Sequence in context: A389983 A319555 A244980 * A367311 A021863 A259620

Adjacent sequences: A394101 A394102 A394103 * A394105 A394106 A394107

Keyword

nonn,cons

Author

Amiram Eldar, Mar 10 2026

Status

approved