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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.
4
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
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
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Oct 29 2021
STATUS
approved