A394919 Decimal expansion of the average length of a chord in a unit cube defined by a point in the interior and a direction, both uniformly and independently chosen at random.

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

Offset

0,1

Links

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

Rodney Coleman, Random paths through convex bodies, Journal of Applied Probability, Vol. 6, No. 2 (1969), pp. 430-441; alternative link; ResearchGate link.

Formula

Equals 1 + log(2)/Pi + (3/2 - 5/(4*Pi))*log(3) - 4*sqrt(2)*arctan(1/sqrt(2))/Pi - (6/Pi) * Integral_{x=0..1} arctan(sqrt(x))/(x+2) dx.

Example

0.896633615389100609722794119741210395458723033374291...

Mathematica

RealDigits[Re[1 + Log[2]/Pi + (3/2 - 5/(4*Pi)) * Log[3] - 4*Sqrt[2]*ArcTan[1/Sqrt[2]]/Pi - (6/Pi) * Integrate[ArcTan[Sqrt[x]]/(x + 2), {x, 0, 1}]], 10, 120][[1]]

Prog

(PARI) 1 + log(2)/Pi + (3/2 - 5/(4*Pi)) * log(3) - 4*sqrt(2)*atan(1/sqrt(2))/Pi - (6/Pi) * intnum(x=0, 1, atan(sqrt(x))/(x+2))

Crossrefs

Cf. A348682, A348683, A394920 (square).

Sequence in context: A048271 A203146 A203128 * A178856 A321161 A386466

Adjacent sequences: A394916 A394917 A394918 * A394920 A394921 A394922

Keyword

nonn,cons

Author

Amiram Eldar, Apr 07 2026

Status

approved