A248589 - OEIS

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

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OFFSET

1,3

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

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.