A394106 Decimal expansion of the probability that three points uniformly and independently selected at random from the interior of a cube form the vertices of an obtuse triangle.

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

Offset

0,1

Links

Table of n, a(n) for n=0..104.

Dominik Beck, The Probability that a Random Triangle in a Cube is Obtuse, arXiv:2501.11611 [math.MG], 2025.

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

Formula

Equals 323338/385875 - 13*G/35 + 4859*Pi/62720 - 73*Pi/(1680*sqrt(2)) - Pi^2/105 + 3*Pi*log(2)/224 - 3*Pi*log(1 + sqrt(2))/224, where G is Catalan's constant (A006752).

Example

0.542659281427229074501111872581772671657167326024954...

Mathematica

RealDigits[323338/385875 - 13*Catalan/35 + 4859*Pi/62720 - 73*Pi/(1680*Sqrt[2]) - Pi^2/105 + 3*Pi*Log[2]/224 - 3*Pi*Log[1 + Sqrt[2]]/224, 10, 120][[1]]

Prog

(PARI) 323338/385875 - 13*Catalan/35 + 4859*Pi/62720 - 73*Pi/(1680*sqrt(2)) - Pi^2/105 + 3*Pi*log(2)/224 - 3*Pi*log(1 + sqrt(2))/224

Crossrefs

Cf. A006752.

Related constants: A075549, A093072, A093588, A102519, A102520, A341942, A394107.

Sequence in context: A260849 A246746 A180131 * A257972 A222307 A368274

Adjacent sequences: A394103 A394104 A394105 * A394107 A394108 A394109

Keyword

nonn,cons

Author

Amiram Eldar, Mar 10 2026

Status

approved