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%I #7 Mar 11 2026 15:06:39
%S 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,
%T 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,
%U 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
%N 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.
%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/Cuboctahedron.html">Cuboctahedron</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Cuboctahedron">Cuboctahedron</a>.
%F 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.
%e 0.0130025156810300487127452824874285413306377954045601...
%t 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]]
%o (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
%Y 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).
%K nonn,cons
%O 0,3
%A _Amiram Eldar_, Mar 11 2026