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
Examples
0.47320100440933855164249820976301198129996056896363...
Links
- Muthu Veerappan Ramalingam, An Expected Value Problem
Crossrefs
Cf. A330646.
Programs
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Mathematica
N[(2 - 2 Sqrt[2])/3/Pi + 2 ArcCsch[1]/Pi, 50]
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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.