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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.

%I A248589 #9 Feb 16 2025 08:33:23
%S A248589 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 A248589 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 A248589 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 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.
%D A248589 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 8.1 Geometric probability constants, p. 481.
%H A248589 G. C. Greubel, <a href="/A248589/b248589.txt">Table of n, a(n) for n = 1..5000</a>
%H A248589 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/SquarePointPicking.html">Square Point Picking</a>
%F A248589 I = sqrt(Pi/8)*(2-integral_{1..infinity} (sqrt(1+s^2)-s)*s^(-3/2) ds).
%F A248589 I = sqrt(Pi/2)*A053004, where A053004 is the arithmetic-geometric mean of 1 and sqrt(2).
%F A248589 I = Pi^(3/2)/(4*A085565), where A085565 is the lemniscate constant A.
%F A248589 I = sqrt(2)*Pi^2/Gamma(1/4)^2.
%e A248589 1.061824136490969662805378287398947131005559644732889212...
%t A248589 RealDigits[Sqrt[2]*Pi^2/Gamma[1/4]^2, 10, 100][[1]]
%o A248589 (PARI) sqrt(2)*Pi^2/gamma(1/4)^2 \\ _G. C. Greubel_, Jun 02 2017
%Y A248589 Cf. A053004, A085565, A096428, A096429.
%K A248589 nonn,cons,easy
%O A248589 1,3
%A A248589 _Jean-François Alcover_, Oct 09 2014