A394919 - OEIS

%I #6 Apr 07 2026 08:44:53

%S 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,

%T 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,

%U 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

%N 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.

%H Rodney Coleman, <a href="https://www.jstor.org/stable/3212012">Random paths through convex bodies</a>, Journal of Applied Probability, Vol. 6, No. 2 (1969), pp. 430-441; <a href="https://doi.org/10.2307/3212012">alternative link</a>; <a href="https://www.researchgate.net/publication/268246373">ResearchGate link</a>.

%F 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.

%e 0.896633615389100609722794119741210395458723033374291...

%t 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]]

%o (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))

%Y Cf. A348682, A348683, A394920 (square).

%K nonn,cons

%O 0,1

%A _Amiram Eldar_, Apr 07 2026