A394174 - OEIS

%I #7 Mar 11 2026 15:06:27

%S 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,

%T 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,

%U 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

%N 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.

%H Dominik Beck, <a href="https://arxiv.org/abs/2412.07952">On Random Simplex Picking Beyond the Blashke Problem</a>, arXiv:2412.07952 [math.MG], 2024.

%H Dominik Beck, <a href="https://www2.karlin.mff.cuni.cz/~beckd/lectures/DISSERTATION.pdf">Random polytopes</a>, doctoral thesis, Mathematical Institute of Charles University, Prague, 2025.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TriangularDipyramid.html">Triangular Dipyramid</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Triangular_bipyramid">Triangular bipyramid</a>.

%F 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.

%e 0.0150824273269304328371358172831191947141991750456702...

%t 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]]

%o (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

%Y 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).

%K nonn,cons

%O 0,3

%A _Amiram Eldar_, Mar 11 2026