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

Author

Muthu Veerappan Ramalingam, Jan 03 2020

Keywords

• nonn
• cons

Examples

			0.47320100440933855164249820976301198129996056896363...
		

Links

• Muthu Veerappan Ramalingam, An Expected Value Problem

Crossrefs

Cf. A330646.

Programs

Mathematica

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

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

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.