cp's OEIS Frontend

A348682 Decimal expansion of the average length of a chord in a unit cube defined by a point on the surface and a direction, both uniformly and independently chosen at random.

%I A348682 #5 Oct 29 2021 09:06:19
%S A348682 5,9,7,7,5,5,7,4,3,5,9,2,7,3,3,7,3,9,8,1,5,1,9,6,0,7,9,8,2,7,4,7,3,5,
%T A348682 9,6,9,7,2,4,8,2,0,2,2,2,4,9,5,2,7,8,5,1,5,6,1,8,2,9,5,0,4,3,2,5,0,3,
%U A348682 8,0,6,5,1,5,0,4,9,6,7,8,2,2,9,3,2,7,4,9,5,1,6,1,5,5,0,3,7,1,0,8,1,4,1,1,0
%N A348682 Decimal expansion of the average length of a chord in a unit cube defined by a point on the surface and a direction, both uniformly and independently chosen at random.
%H A348682 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_Random_Paths_Through_Convex_Bodies">author's link</a>.
%H A348682 Maurice Horowitz, <a href="https://www.jstor.org/stable/3211882">Probability of random paths across elementary geometrical shapes</a>, Journal of Applied Probability, Vol. 2, No. 1 (1965), pp. 169-177; <a href="https://www.jstor.org/stable/3212055">Correction</a>, ibid., Vol. 3, No. 1 (1966), p. 285.
%F A348682 Equals (1/(3*Pi)) * (2*Pi - 6 + 2*log(2) + 7*log(3)/2 + 4*sqrt(2)*arccot(sqrt(2))) - (4/Pi) * Integral_{x=1..sqrt(2)} (sqrt(x^2-1) * (x * arccot(x) + log(1 + x^2)/2) / x) dx.
%e A348682 0.5977557435927337398151960798274735969724820222495278516...
%t A348682 RealDigits[N[(1/(3*Pi)) * (2*Pi - 6 + 2*Log[2] + 7*Log[3]/2 + 4*Sqrt[2]*ArcCot[Sqrt[2]]) - (4/Pi) * Integrate[Sqrt[x^2-1] * (x * ArcCot[x] + Log[1 + x^2]/2) / x, {x, 1, Sqrt[2]}], 110], 10, 100][[1]]
%Y A348682 Cf. A073012, A093066, A348680, A348681, A348683.
%K A348682 nonn,cons
%O A348682 0,1
%A A348682 _Amiram Eldar_, Oct 29 2021