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%I #24 Aug 08 2020 11:29:58
%S 7,6,5,1,9,5,7,1,6,4,6,4,2,1,2,6,9,1,3,4,4,7,6,6,0,1,6,3,9,6,4,9,6,7,
%T 9,5,8,6,5,9,4,4,0,6,7,8,7,9,5,2,7,8,2,7,9,7,6,6,5,8,4,4,8,8,8,1,3,6,
%U 9,8,8,7,5,6,1,3,7,7,7,0,8,8,9,4,6,9,8,1,4,2,0,7,9,2,9,9,2,0,5,1,9,7,2,5
%N Decimal expansion of (sqrt(2)+log(1+sqrt(2)))/3, the integral over the square [0,1]x[0,1] of sqrt(x^2+y^2) dx dy.
%C This is also the expected distance from a randomly selected point in the unit square to a corner, as well as the expected distance from a randomly selected point in a 45-45-90 degree triangle of base length 1 to a vertex with an acute angle. - _Derek Orr_, Jul 27 2014
%C The average length of chords in a unit square drawn between two points uniformly and independently chosen at random on two adjacent sides. - _Amiram Eldar_, Aug 08 2020
%H Vincenzo Librandi, <a href="/A244921/b244921.txt">Table of n, a(n) for n = 0..1000</a>
%H D. Bailey, J. Borwein, and R. Crandall, <a href="https://doi.org/10.1090/S0025-5718-10-02338-0">Advances in the theory of box integrals</a>, Mathematics of Computation, Vol. 79, No. 271 (2010), pp. 1839-1866. See p. 1860.
%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.
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F Also equals (sqrt(2) + arcsinh(1))/3.
%F This is also 2*A103712. - _Derek Orr_, Jul 27 2014
%e 0.76519571646421269134476601639649679586594406787952782797665844888136988756...
%t RealDigits[(Sqrt[2] + Log[1 + Sqrt[2]])/3, 10, 104] // First
%o (PARI) (sqrt(2)+log(1+sqrt(2)))/3 \\ _G. C. Greubel_, Jul 05 2017
%Y Cf. A244920.
%K nonn,cons,easy
%O 0,1
%A _Jean-François Alcover_, Jul 08 2014
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