A394174 Decimal expansion of the mean volume of a tetrahedron whose vertices are uniformly and independently selected at random in the interior of a triangular bipyramid (or dipyramid) of unit volume.

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

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, Triangular Dipyramid.

Wikipedia, Triangular bipyramid.

Formula

Equals 1712190037/16812956160 + 81471636487*Pi^2/907899632640 - 185777703053*log(2)/50438868480 - 909434448983*log(2)^2/121053284352 + 3498264683*log(3)/2401850880 + 20912895*log(2)*log(3)/2050048 - 1887867*log(3)^2/585728 - 62045573287*LolyLog(2, 1/4)/57644421120.

Example

0.0150824273269304328371358172831191947141991750456702...

Mathematica

RealDigits[1712190037/16812956160 + 81471636487*Pi^2/907899632640 - 185777703053*Log[2]/50438868480 - 909434448983*Log[2]^2/121053284352 + 3498264683*Log[3]/2401850880 + 20912895*Log[2]*Log[3]/2050048 - 1887867*Log[3]^2/585728 - 62045573287*PolyLog[2, 1/4]/57644421120, 10, 120, -1][[1]]

Prog

(PARI) 1712190037/16812956160 + 81471636487*Pi^2/907899632640 - 185777703053*log(2)/50438868480 - 909434448983*log(2)^2/121053284352 + 3498264683*log(3)/2401850880 + 20912895*log(2)*log(3)/2050048 - 1887867*log(3)^2/585728 - 62045573287*polylog(2, 1/4)/57644421120

Crossrefs

Tetrahedron volume in: A093524 (cube), A093525 (tetrahedron), A093591 (ball), A394169 (surface of sphere), A394170 (octahedron), A394171 (rhombic dodecahedron), A394172 (cuboctahedron), A394173 (truncated tetrahedron), this constant (triangular bipyramid), A394175 (triangular prism), A394176 (square pyramid).

Sequence in context: A124914 A199376 A011510 * A021667 A078119 A085998

Adjacent sequences: A394171 A394172 A394173 * A394175 A394176 A394177

Keyword

nonn,cons

Author

Amiram Eldar, Mar 11 2026

Status

approved