A394173 Decimal expansion of the mean volume of a tetrahedron whose vertices are uniformly and independently selected at random in the interior of a truncated tetrahedron of unit volume.

0, 1, 4, 8, 4, 5, 1, 0, 1, 9, 0, 2, 8, 9, 5, 3, 7, 3, 0, 7, 2, 1, 7, 6, 2, 1, 1, 9, 0, 5, 0, 1, 2, 5, 1, 4, 0, 4, 6, 6, 4, 5, 2, 5, 1, 8, 9, 3, 9, 3, 0, 4, 5, 6, 6, 9, 1, 0, 3, 1, 9, 8, 2, 6, 1, 1, 7, 3, 8, 6, 4, 4, 3, 0, 1, 9, 0, 9, 6, 9, 1, 9, 2, 3, 1, 3, 0, 0, 3, 4, 0, 9, 0, 2, 4, 6, 2, 0, 4, 1, 5, 5, 9, 1, 6, 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, Truncated Tetrahedron.

Wikipedia, Truncated tetrahedron.

Formula

Equals 35604506258521/162358039443600 - 13447020779*Pi^2/96641690145 + 9972537226592*log(2)/3382459155075 + 3485442712*log(2)^2/1400604205 - 8953623027*log(3)/7884520175 - 53493528168*log(2)*log(3)/32213896715 + 53162662164*PolyLog(2, 1/4)/32213896715.

Example

0.0148451019028953730721762119050125140466452518939304...

Mathematica

RealDigits[35604506258521/162358039443600 - 13447020779*Pi^2/96641690145 + 9972537226592*Log[2]/3382459155075 + 3485442712*Log[2]^2/1400604205 - 8953623027*Log[3]/7884520175 - 53493528168*Log[2]*Log[3]/32213896715 + 53162662164*PolyLog[2, 1/4]/32213896715, 10, 120, -1][[1]]

Prog

(PARI) 35604506258521/162358039443600 - 13447020779*Pi^2/96641690145 + 9972537226592*log(2)/3382459155075 + 3485442712*log(2)^2/1400604205 - 8953623027*log(3)/7884520175 - 53493528168*log(2)*log(3)/32213896715 + 53162662164*polylog(2, 1/4)/32213896715

Crossrefs

Tetrahedron volume in: A093524 (cube), A093525 (tetrahedron), A093591 (ball), A394169 (surface of sphere), A394170 (octahedron), A394171 (rhombic dodecahedron), A394172 (cuboctahedron), this constant (truncated tetrahedron), A394174 (triangular bipyramid), A394175 (triangular prism), A394176 (square pyramid).

Sequence in context: A045816 A085991 A122110 * A082632 A296481 A155874

Adjacent sequences: A394170 A394171 A394172 * A394174 A394175 A394176

Keyword

nonn,cons

Author

Amiram Eldar, Mar 11 2026

Status

approved