A249491 Decimal expansion of the expected product of two sides of a random Gaussian triangle (in two dimensions).

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, 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, 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

Offset: 1

Author

Jean-François Alcover, Oct 30 2014

Keywords

• nonn
• cons
• easy

Comments

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

Examples

			3.341223305138814557532375581265049059850245668...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000
• Steven R. Finch, Random Triangles, Jan 21 2010. [Cached copy, with permission of the author]
• Eric Weisstein's World of Mathematics, Gaussian Triangle Picking

Crossrefs

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

Programs

Maple

Re(evalf(4*EllipticE(1/2)-sqrt(3)*EllipticK(I/sqrt(3)), 120)); # Vaclav Kotesovec, Apr 22 2015

Mathematica

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
RealDigits[ N[ EllipticE[-8], 102]][[1]] (* Altug Alkan, Oct 02 2018 *)
RealDigits[3 EllipticE[8/9], 10, 102][[1]] (* Jan Mangaldan, Nov 24 2020 *)

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
E(x)=Pi/2/agm(1,sqrt(1-x))*magm(1,1-x)
K(x)=Pi/2/agm(1,sqrt(1-x))
4*E(1/4) - sqrt(3)*K(-1/3) \\ Charles R Greathouse IV, Aug 02 2018

Formula

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.

Equals A093728/2. - Altug Alkan, Oct 02 2018