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

1,1

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = 1..5000

S. R. Finch, Random Triangles, Jan 21 2010. [Cached copy, with permission of the author]

Eric Weisstein MathWorld, Gaussian Triangle Picking

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

Example

3.341223305138814557532375581265049059850245668...

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A078911 A353417 A082899 * A309888 A245250 A179561

Adjacent sequences: A249488 A249489 A249490 * A249492 A249493 A249494

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Oct 30 2014

Status

approved