A330924 Decimal expansion of the mean length of a line segment drawn by picking a random point on a unit square and extending it in a random direction until it meets an edge.

4, 7, 3, 2, 0, 1, 0, 0, 4, 4, 0, 9, 3, 3, 8, 5, 5, 1, 6, 4, 2, 4, 9, 8, 2, 0, 9, 7, 6, 3, 0, 1, 1, 9, 8, 1, 2, 9, 9, 9, 6, 0, 5, 6, 8, 9, 6, 3, 6, 3, 4, 9, 8, 2, 3, 7, 2, 6, 3, 6, 6, 1, 4, 3, 3, 8, 5, 9, 8, 6, 7, 5, 7, 2, 7, 3, 4, 2, 9, 6, 1, 5, 0, 8, 0, 6, 9, 5, 3, 2, 5, 5, 3, 0, 4, 6, 7, 5, 0, 5, 3, 1, 9, 4, 1, 0, 5, 5

Offset

0,1

Links

Table of n, a(n) for n=0..107.

Muthu Veerappan Ramalingam, An Expected Value Problem

Formula

Equals (2 - 2 * sqrt(2)) / (3 * Pi) + (2 / Pi) * cosech^{-1}(1)

Equals (a^3 + b^3 - d^3) / (3 * Pi * a * b) + (a / Pi) * cosech^{-1}(a / b) + (b / Pi) * cosech^{-1}(b / a) for an arbitrary rectangle with sides a, b and diagonal d.

Example

0.47320100440933855164249820976301198129996056896363...

Mathematica

N[(2 - 2 Sqrt[2])/3/Pi + 2 ArcCsch[1]/Pi, 50]

Prog

(PARI) arcsch(x) = log(1/x + sqrt(1+1/x^2));
(2 - 2*sqrt(2))/(3*Pi) + (2/ Pi)*arcsch(1) \\ Michel Marcus, Jan 16 2020

Crossrefs

Cf. A330646.

Sequence in context: A200602 A118823 A118826 * A165663 A254397 A197697

Adjacent sequences: A330921 A330922 A330923 * A330925 A330926 A330927

Keyword

nonn,cons

Author

Muthu Veerappan Ramalingam, Jan 03 2020

Status

approved