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A249492 Decimal expansion of rho(a,b), the cross-correlation coefficient of two sides of a random Gaussian triangle (in two dimensions).

%I A249492 #20 Feb 16 2025 08:33:24
%S A249492 2,3,2,5,5,9,3,4,6,5,4,3,1,7,8,2,3,4,4,7,3,0,9,0,3,5,9,7,5,0,3,3,3,8,
%T A249492 9,9,3,1,0,4,3,5,0,1,5,4,3,5,0,2,0,4,0,9,8,8,5,9,9,4,2,1,0,5,9,7,7,6,
%U A249492 1,7,9,9,9,1,4,9,8,0,9,1,9,1,7,5,9,5,4,5,1,2,5,4,6,9,0,8,3,8,5,2,7,8,4
%N A249492 Decimal expansion of rho(a,b), the cross-correlation coefficient of two sides of a random Gaussian triangle (in two dimensions).
%C A249492 Coordinates are independent normally distributed random variables with mean 0 and variance 1.
%H A249492 G. C. Greubel, <a href="/A249492/b249492.txt">Table of n, a(n) for n = 0..5000</a>
%H A249492 Steven R. Finch, <a href="/A102519/a102519.pdf">Random Triangles</a>, January 21, 2010. [Cached copy, with permission of the author]
%H A249492 Eric Weisstein MathWorld, <a href="https://mathworld.wolfram.com/GaussianTrianglePicking.html">Gaussian Triangle Picking</a>
%F A249492 rho = (p - Pi)/(4 - Pi), where p is A249491, the expected value of the product of two sides.
%e A249492 0.23255934654317823447309035975033389931043501543502...
%p A249492 Re(evalf((4*EllipticE(1/2) - sqrt(3)*EllipticK(I/sqrt(3)) - Pi)/(4 - Pi), 120)); # _Vaclav Kotesovec_, Apr 22 2015
%t A249492 p = 4*EllipticE[1/4] - Sqrt[3]*EllipticK[-1/3]; rho = (p - Pi)/(4 - Pi); RealDigits[rho, 10, 103] // First
%t A249492 RealDigits[(3 EllipticE[8/9] - Pi)/(4 - Pi), 10, 103][[1]] (* _Jan Mangaldan_, Nov 26 2020 *)
%Y A249492 Cf. A102519, A102520, A102556, A102557, A102558, A102559, A249491.
%K A249492 nonn,cons,easy
%O A249492 0,1
%A A249492 _Jean-François Alcover_, Oct 30 2014