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%I #20 Sep 08 2022 08:46:09
%S 1,0,7,6,6,3,5,7,3,2,8,9,5,1,7,8,0,0,8,9,6,5,3,7,9,7,5,0,2,4,3,2,2,6,
%T 2,8,2,8,3,8,2,6,9,7,0,3,1,3,5,9,8,6,0,5,3,0,2,7,7,3,5,6,9,5,9,8,9,7,
%U 9,9,6,9,1,4,0,1,3,2,3,7,4,1,5,5,0,2,4,4,3,8,0,4,6,7,7,0,8,8,5,1,9,4,5
%N Decimal expansion of the integral over the square [0,1]x[0,1] of sqrt(1+(x-y)^2) dx dy.
%C The average length of chords in a unit square drawn between two points uniformly and independently chosen at random on two opposite sides. - _Amiram Eldar_, Aug 08 2020
%H G. C. Greubel, <a href="/A247674/b247674.txt">Table of n, a(n) for n = 1..10000</a>
%H D. H. Bailey and J. M. Borwein, <a href="https://escholarship.org/uc/item/4281090t">Highly Parallel, High-Precision Numerical Integration</a>, Lawrence Berkeley National Laboratory (2005), p. 9.
%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 2/3 - sqrt(2)/3 + arcsinh(1).
%F Equals 2*A244921 + A247674 = (2 + sqrt(2) + 5*log(1+sqrt(2)))/3.
%e 1.076635732895178008965379750243226282838269703135986...
%t RealDigits[2/3 - Sqrt[2]/3 + ArcSinh[1], 10, 103] // First
%o (PARI) default(realprecision, 100); (2 + sqrt(2) + 5*log(1+sqrt(2)))/3 \\ _G. C. Greubel_, Aug 31 2018
%o (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); (2 + Sqrt(2) + 5*Log(1+Sqrt(2)))/3; // _G. C. Greubel_, Aug 31 2018
%Y Cf. A244921.
%K nonn,cons
%O 1,3
%A _Jean-François Alcover_, Sep 22 2014
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