%I #25 Apr 25 2021 01:37:46

%S 1,6,3,7,7,2,9,3,2,4,8,5,6,8,6,8,0,3,2,7,8,0,1,5,6,9,5,6,7,9,8,4,7,6,

%T 4,5,5,8,2,0,3,8,1,9,8,7,0,9,0,5,9,3,4,1,7,5,4,8,7,6,5,2,2,4,7,7,1,2,

%U 0,5,6,8,9,3,3,1,1,1,6,4,9,0,2,1,5,0,7,1,1,3,4,8,3,2,2,0,7,1,2,4,6,9,9,2,8

%N Decimal expansion of the average product of a side and an adjacent angle of a random Gaussian triangle in two dimensions.

%C Coordinates are independent normally distributed random variables with mean 0 and variance 1.

%C As of 2010, an exact expression of this constant was not known, according to Steven Finch.

%C This average product is noticeably smaller than the product of the averages sqrt(Pi)*Pi/3 = 1.8561..., the side length being negatively correlated with the adjacent angle value.

%H G. C. Greubel, <a href="/A249542/b249542.txt">Table of n, a(n) for n = 1..5000</a>

%H Steven R. Finch, <a href="/A102519/a102519.pdf">Random Triangles</a>, January 21, 2010, p. 14. [Cached copy, with permission of the author]

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/GaussianTrianglePicking.html">Gaussian Triangle Picking</a>

%F Equals (1/(3*Pi)*Integral_{t=0..infinity} (Integral_{y=0..infinity} (Integral_{t=0..Pi} x^2*y*t*exp(-(1/3)*x^2-x*y*cos(t) + y^2) dt) dy) dx.

%F Equals (-sqrt(3)*log(3) + Pi^2 - 8*Li_2(2-sqrt(3)) + 2*Li_2(7-4*sqrt(3)))/(2*sqrt(Pi)), where Li_2 is the dilogarithm function.

%e 1.6377293248568680327801569567984764558203819870905934...

%t ex = (-Sqrt[3]*Log[3] + Pi^2 - 8*PolyLog[2, 2-Sqrt[3]] + 2*PolyLog[2, 7-4*Sqrt[3]])/(2*Sqrt[Pi]); RealDigits[ex, 10, 105] // First

%o (Python)

%o from mpmath import *

%o mp.dps=106

%o C = (-sqrt(3)*log(3) + pi**2 - 8*polylog(2, 2-sqrt(3)) + 2*polylog(2, 7 - 4*sqrt(3)))/(2*sqrt(pi))

%o print([int(n) for n in list(str(C).replace('.', '')[:-1])]) # _Indranil Ghosh_, Jul 04 2017

%Y Cf. A102519, A102520, A102556, A102557, A102558, A102559, A249491, A249492, A249538.

%K nonn,cons

%O 1,2

%A _Jean-François Alcover_, Oct 31 2014