login
A248589
Decimal expansion of I, a constant appearing (as I^2) in the asymptotic variance of the area of the convex hull of random points in the unit square.
1
1, 0, 6, 1, 8, 2, 4, 1, 3, 6, 4, 9, 0, 9, 6, 9, 6, 6, 2, 8, 0, 5, 3, 7, 8, 2, 8, 7, 3, 9, 8, 9, 4, 7, 1, 3, 1, 0, 0, 5, 5, 5, 9, 6, 4, 4, 7, 3, 2, 8, 8, 9, 2, 1, 2, 0, 4, 0, 5, 0, 1, 5, 1, 8, 3, 3, 8, 9, 8, 3, 3, 4, 5, 5, 6, 1, 2, 1, 1, 6, 1, 2, 4, 1, 3, 6, 9, 0, 0, 1, 0, 4, 2, 5, 9, 4, 5, 9, 0, 2
OFFSET
1,3
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 8.1 Geometric probability constants, p. 481.
LINKS
Eric Weisstein's MathWorld, Square Point Picking
FORMULA
I = sqrt(Pi/8)*(2-integral_{1..infinity} (sqrt(1+s^2)-s)*s^(-3/2) ds).
I = sqrt(Pi/2)*A053004, where A053004 is the arithmetic-geometric mean of 1 and sqrt(2).
I = Pi^(3/2)/(4*A085565), where A085565 is the lemniscate constant A.
I = sqrt(2)*Pi^2/Gamma(1/4)^2.
EXAMPLE
1.061824136490969662805378287398947131005559644732889212...
MATHEMATICA
RealDigits[Sqrt[2]*Pi^2/Gamma[1/4]^2, 10, 100][[1]]
PROG
(PARI) sqrt(2)*Pi^2/gamma(1/4)^2 \\ G. C. Greubel, Jun 02 2017
CROSSREFS
KEYWORD
nonn,cons,easy
AUTHOR
STATUS
approved