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%I #7 Oct 29 2021 09:05:44
%S 7,0,9,8,0,1,5,0,6,6,1,4,0,0,7,8,2,7,4,6,3,7,4,7,3,1,4,6,4,4,5,1,7,9,
%T 7,1,9,4,9,9,4,0,8,5,3,4,4,5,4,5,2,4,7,3,5,5,8,9,5,4,9,2,1,5,0,7,8,9,
%U 8,0,1,3,5,9,1,0,1,4,4,4,2,2,6,2,1,0,4,2,9,8,8,2,9,5,7,0,1,2,5,7,9,7,9,1,1
%N Decimal expansion of the average length of a chord in a unit square defined by a point on the perimeter and a direction, both uniformly and independently chosen at random.
%D A. M. Mathai, An introduction to geometrical probability: distributional aspects with applications, Amsterdam: Gordon and Breach, 1999, p. 221, ex. 2.3.7.
%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_Random_Paths_Through_Convex_Bodies">author's link</a>.
%H 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.
%H Philip W. Kuchel and Rodney J. Vaughan, <a href="https://www.jstor.org/stable/2689989">Average Lengths of Chords in a Square</a>, Mathematics Magazine, Vol. 54, No. 5 (1981), pp. 261-269.
%F Equals (3*log(1 + sqrt(2)) + 1 - sqrt(2))/Pi.
%e 0.70980150661400782746374731464451797194994085344545...
%t RealDigits[(3 * Log[1 + Sqrt[2]] + 1 - Sqrt[2])/Pi, 10, 100][[1]]
%o (PARI) (3*log(1 + sqrt(2)) + 1 - sqrt(2))/Pi \\ _Michel Marcus_, Oct 29 2021
%Y Cf. A091505, A247674, A348681, A348682, A348683.
%K nonn,cons
%O 0,1
%A _Amiram Eldar_, Oct 29 2021
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