This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A249542 #30 Feb 28 2025 05:54:06
%S A249542 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 A249542 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 A249542 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 A249542 Decimal expansion of the average product of a side and an adjacent angle of a random Gaussian triangle in two dimensions.
%C A249542 Coordinates are independent normally distributed random variables with mean 0 and variance 1.
%C A249542 As of 2010, an exact expression of this constant was not known, according to Steven Finch.
%C A249542 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 A249542 G. C. Greubel, <a href="/A249542/b249542.txt">Table of n, a(n) for n = 1..5000</a>
%H A249542 Steven R. Finch, <a href="/A102519/a102519.pdf">Random Triangles</a>, January 21, 2010, p. 14. [Cached copy, with permission of the author]
%H A249542 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/GaussianTrianglePicking.html">Gaussian Triangle Picking</a>
%F A249542 Equals (1/(3*Pi))*Integral_{x=0..oo} Integral_{y=0..oo} Integral_{t=0..Pi} x^2*y*t*exp(-(1/3)*(x^2 - x*y*cos(t) + y^2)) dt dy dx.
%F A249542 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 A249542 1.6377293248568680327801569567984764558203819870905934...
%t A249542 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 A249542 (Python)
%o A249542 from mpmath import *
%o A249542 mp.dps=106
%o A249542 C = (-sqrt(3)*log(3) + pi**2 - 8*polylog(2, 2-sqrt(3)) + 2*polylog(2, 7 - 4*sqrt(3)))/(2*sqrt(pi))
%o A249542 print([int(n) for n in list(str(C).replace('.', '')[:-1])]) # _Indranil Ghosh_, Jul 04 2017
%Y A249542 Cf. A102519, A102520, A102556, A102557, A102558, A102559, A249491, A249492, A249538.
%K A249542 nonn,cons
%O A249542 1,2
%A A249542 _Jean-François Alcover_, Oct 31 2014