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

1,3

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.

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

Cf. A053004, A085565, A096428, A096429.

Sequence in context: A349905 A258404 A244692 * A288493 A195293 A230763

Adjacent sequences: A248586 A248587 A248588 * A248590 A248591 A248592

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Oct 09 2014

Status

approved