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.

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, 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, 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

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; author's link.

Maurice Horowitz, Probability of random paths across elementary geometrical shapes, Journal of Applied Probability, Vol. 2, No. 1 (1965), pp. 169-177; Correction, ibid., Vol. 3, No. 1 (1966), p. 285.

Formula

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.

Example

0.5977557435927337398151960798274735969724820222495278516...

Mathematica

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]]

Crossrefs

Cf. A073012, A093066, A348680, A348681, A348683.

Sequence in context: A220261 A365238 A195285 * A200597 A140724 A086055

Adjacent sequences: A348679 A348680 A348681 * A348683 A348684 A348685

Keyword

nonn,cons

Author

Amiram Eldar, Oct 29 2021

Status

approved