A394172 Decimal expansion of the mean volume of a tetrahedron whose vertices are uniformly and independently selected at random in the interior of a cuboctahedron of unit volume.

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

Offset

0,3

Links

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

Dominik Beck, On Random Simplex Picking Beyond the Blashke Problem, arXiv:2412.07952 [math.MG], 2024.

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

Eric Weisstein's World of Mathematics, Cuboctahedron.

Wikipedia, Cuboctahedron.

Formula

Equals 117410162173/525525000000 + 8752199*Pi^2/2402400000 - 192940695481*log(2)/105105000000 - 318759601*log(2)^2/250250000 + 506316394917*log(3)/280280000000 - 648098487*PolyLog(2, 1/4)/500500000.

Example

0.0130025156810300487127452824874285413306377954045601...

Mathematica

RealDigits[117410162173/525525000000 + 8752199*Pi^2/2402400000 - 192940695481*Log[2]/105105000000 - 318759601*Log[2]^2/250250000 + 506316394917*Log[3]/280280000000 - 648098487*PolyLog[2, 1/4]/500500000, 10, 120, -1][[1]]

Prog

(PARI) 117410162173/525525000000 + 8752199*Pi^2/2402400000 - 192940695481*log(2)/105105000000 - 318759601*log(2)^2/250250000 + 506316394917*log(3)/280280000000 - 648098487*polylog(2, 1/4)/500500000

Crossrefs

Tetrahedron volume in: A093524 (cube), A093525 (tetrahedron), A093591 (ball), A394169 (surface of sphere), A394170 (octahedron), A394171 (rhombic dodecahedron), this constant (cuboctahedron), A394173 (truncated tetrahedron), A394174 (triangular bipyramid), A394175 (triangular prism), A394176 (square pyramid).

Sequence in context: A357236 A156548 A112883 * A117138 A292255 A362313

Adjacent sequences: A394169 A394170 A394171 * A394173 A394174 A394175

Keyword

nonn,cons

Author

Amiram Eldar, Mar 11 2026

Status

approved