A330924 - OEIS

%I #14 Jan 16 2020 13:18:44

%S 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,

%T 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,

%U 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

%N 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.

%H Muthu Veerappan Ramalingam, <a href="https://am-just-a-nobody.blogspot.com/2017/06/an-expected-value-problem.html">An Expected Value Problem</a>

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

%F 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.

%e 0.47320100440933855164249820976301198129996056896363...

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

%o (PARI) arcsch(x) = log(1/x + sqrt(1+1/x^2));

%o (2 - 2*sqrt(2))/(3*Pi) + (2/ Pi)*arcsch(1) \\ _Michel Marcus_, Jan 16 2020

%Y Cf. A330646.

%K nonn,cons

%O 0,1

%A _Muthu Veerappan Ramalingam_, Jan 03 2020