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%I #25 Sep 08 2022 08:46:09
%S 5,4,1,0,7,5,0,8,0,0,4,6,7,4,3,5,0,4,4,6,4,6,7,3,3,6,0,0,8,3,5,2,2,6,
%T 6,7,5,5,0,2,3,1,7,7,0,7,8,2,1,8,9,0,8,4,2,9,9,5,7,1,5,9,2,0,3,2,0,5,
%U 6,6,6,8,1,8,2,3,3,8,0,6,0,1,5,5,8,8,9,6,9,1,0,7,8,5,4,2,2,0,9,3,5,6,5,2,7,8,8,4,0,3,0,4,7,4,2,3,1,8,1,4
%N Decimal expansion of the expected distance from a randomly selected point in a 45-45-90 degree triangle of base length 1 to the vertex of the right angle: (4+sqrt(2)*log(3+2*sqrt(2)))/12.
%H G. C. Greubel, <a href="/A245699/b245699.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F Equals Integral_{y = 0..Pi/4; x = 0..1/(sqrt(2)*cos(y))} 4x^2 dx dy.
%F Equals Integral_{y = 0..Pi/4} (sqrt(2)/3)*sec^3(y) dy.
%e 0.54107508004674350446467336008352266755023177078218908429957159203205...
%p evalf((4+sqrt(2)*log(3+2*sqrt(2)))/12,100); # _Muniru A Asiru_, Oct 07 2018
%t RealDigits[(4 + Sqrt[2]*Log[3 + 2*Sqrt[2]])/12, 10, 100][[1]] (* _G. C. Greubel_, Oct 06 2018 *)
%o (PARI) default(realprecision, 100); (4+sqrt(2)*log(3+2*sqrt(2)))/12 \\ _G. C. Greubel_, Oct 06 2018
%o (Magma) SetDefaultRealField(RealField(100)); (4+Sqrt(2)*Log(3 +2*Sqrt(2)))/12; // _G. C. Greubel_, Oct 06 2018
%Y Cf. A103712.
%K nonn,cons
%O 0,1
%A _Derek Orr_, Jul 29 2014
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