A248589 - OEIS

%I #8 Jun 02 2017 07:03:22

%S 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,

%T 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,

%U 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

%N 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.

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

%H G. C. Greubel, <a href="/A248589/b248589.txt">Table of n, a(n) for n = 1..5000</a>

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/SquarePointPicking.html">Square Point Picking</a>

%F I = sqrt(Pi/8)*(2-integral_{1..infinity} (sqrt(1+s^2)-s)*s^(-3/2) ds).

%F I = sqrt(Pi/2)*A053004, where A053004 is the arithmetic-geometric mean of 1 and sqrt(2).

%F I = Pi^(3/2)/(4*A085565), where A085565 is the lemniscate constant A.

%F I = sqrt(2)*Pi^2/Gamma(1/4)^2.

%e 1.061824136490969662805378287398947131005559644732889212...

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

%o (PARI) sqrt(2)*Pi^2/gamma(1/4)^2 \\ _G. C. Greubel_, Jun 02 2017

%Y Cf. A053004, A085565, A096428, A096429.

%K nonn,cons,easy

%O 1,3

%A _Jean-François Alcover_, Oct 09 2014