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A249491 Decimal expansion of the expected product of two sides of a random Gaussian triangle (in two dimensions). 6
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Coordinates are independent normally distributed random variables with mean 0 and variance 1.
LINKS
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
Sequence in context: A078911 A353417 A082899 * A309888 A245250 A179561
KEYWORD
nonn,cons,easy
AUTHOR
STATUS
approved

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Last modified December 8 11:46 EST 2023. Contains 367678 sequences. (Running on oeis4.)