%I #31 Sep 08 2022 08:45:12
%S 8,6,9,0,0,9,0,5,5,2,7,4,5,3,4,4,6,3,8,8,4,9,7,0,5,9,4,3,4,5,4,0,6,6,
%T 2,4,8,5,6,7,1,9,2,7,9,6,3,1,6,8,0,5,6,9,6,6,0,3,5,0,8,6,4,5,8,4,1,7,
%U 9,8,2,2,1,7,4,6,9,3,0,5,3,1,1,3,2,1,3,5,5,4,8,7,5,4,3,5,7,5,4,1,1,3
%N Decimal expansion of (2 + sqrt(2) + 5*arcsinh(1))/9.
%C Average distance between two points chosen at random on two different edges of a unit square.
%D Jonathan Borwein, David Bailey, and Roland Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery. Natick: A K Peters (2004): p. 66, Example 57(a).
%H Vincenzo Librandi, <a href="/A091506/b091506.txt">Table of n, a(n) for n = 0..1000</a>
%H D. H. Bailey, J. M. Borwein, V. Kapoor and E. Weisstein, <a href="http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/tenproblems.pdf">Ten Problems in Experimental Mathematics</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SquareLinePicking.html">Square Line Picking</a>
%F Also equals (2 + sqrt(2) + 5*log(1 + sqrt(2)))/9. - _Jean-François Alcover_, Feb 14 2014
%F James D. Klein proved that this constant is equal to 2/3*int(int(sqrt(x^2 + y^2), x = 0..1), y = 0..1) + 1/3*int(int(sqrt(1 + (y - u)^2), u = 0..1), y = 0..1). - _John M. Campbell_, Apr 02 2016
%e 0.86900905527453446388497059434540662485671927963168056966035...
%t RealDigits[(2 + Sqrt[2] + 5ArcSinh[1])/9, 10, 120][[1]] (* _Harvey P. Dale_, May 22 2013 *)
%o (PARI) (2 + sqrt(2) + 5*asinh(1))/9 \\ _G. C. Greubel_, Aug 17 2018
%o (Magma) SetDefaultRealField(RealField(100)); (2 + Sqrt(2) + 5*Argsinh(1))/9; // _G. C. Greubel_, Aug 17 2018
%Y Cf. A091505.
%K nonn,easy,cons
%O 0,1
%A _Eric W. Weisstein_, Jan 16 2004
%E Broken link fixed by _John M. Campbell_, Apr 02 2016
|