A249491 - OEIS

%I #26 Nov 26 2020 11:03:24

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

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

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

%N Decimal expansion of the expected product of two sides of a random Gaussian triangle (in two dimensions).

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

%H G. C. Greubel, <a href="/A249491/b249491.txt">Table of n, a(n) for n = 1..5000</a>

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

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

%F p = 4*E(1/4) - sqrt(3)*K(-1/3), where E is the complete elliptic integral and K the complete elliptic integral of the first kind.

%F Equals A093728/2. - _Altug Alkan_, Oct 02 2018

%e 3.341223305138814557532375581265049059850245668...

%p Re(evalf(4*EllipticE(1/2)-sqrt(3)*EllipticK(I/sqrt(3)), 120)); # _Vaclav Kotesovec_, Apr 22 2015

%t ek[x_] := EllipticK[x^2/(-1 + x^2)]/Sqrt[1 - x^2]; ee[x_] := EllipticE[x^2]; p = 4*ee[1/2] - (3/2)*ek[1/2]; (* or *) p = 4*EllipticE[1/4] - Sqrt[3]*EllipticK[-1/3]; RealDigits[p, 10, 104] // First

%t RealDigits[ N[ EllipticE[-8], 102]][[1]] (* _Altug Alkan_, Oct 02 2018 *)

%t RealDigits[3 EllipticE[8/9], 10, 102][[1]] (* _Jan Mangaldan_, Nov 24 2020 *)

%o (PARI) magm(a,b)=my(eps=10^-(default(realprecision)-5), c); while(abs(a-b)>eps, my(z=sqrt((a-c)*(b-c))); [a,b,c] = [(a+b)/2,c+z,c-z]); (a+b)/2

%o E(x)=Pi/2/agm(1,sqrt(1-x))*magm(1,1-x)

%o K(x)=Pi/2/agm(1,sqrt(1-x))

%o 4*E(1/4) - sqrt(3)*K(-1/3) \\ _Charles R Greathouse IV_, Aug 02 2018

%Y Cf. A093728, A102519, A102520, A102556, A102557, A102558, A102559.

%K nonn,cons,easy

%O 1,1

%A _Jean-François Alcover_, Oct 30 2014