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, 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

Author

Jean-François Alcover, Oct 09 2014

Keywords

• nonn
• cons
• easy

Examples

			1.061824136490969662805378287398947131005559644732889212...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 8.1 Geometric probability constants, p. 481.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000
• Eric Weisstein's MathWorld, Square Point Picking

Crossrefs

Cf. A053004, A085565, A096428, A096429.

Programs

Mathematica

RealDigits[Sqrt[2]*Pi^2/Gamma[1/4]^2, 10, 100][[1]]

PARI

sqrt(2)*Pi^2/gamma(1/4)^2 \\ G. C. Greubel, Jun 02 2017

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.