A395598 - OEIS

A395598

Decimal expansion of the aspect ratio of a rectangle with the property that a straight line drawn through a uniformly random point on one of the two larger sides, with direction chosen uniformly at random, intersects the other larger side with probability 1/2.

%I #5 May 04 2026 00:10:21

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

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

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

%N Decimal expansion of the aspect ratio of a rectangle with the property that a straight line drawn through a uniformly random point on one of the two larger sides, with direction chosen uniformly at random, intersects the other larger side with probability 1/2.

%C The unique real solution to the equation 2*arctan(x) - log(1 + x^2)/x = Pi/2.

%e 2.445828316431492046186470886388920883276945104598903...

%t RealDigits[r /. FindRoot[2*ArcTan[r] - Log[1 + r^2]/r == Pi/2, {r, 2}, WorkingPrecision -> 120]][[1]]

%o (PARI) solve(r = 2, 3, 2*atan(r) - log(1 + r^2)/r - Pi/2)

%Y Cf. A156649, A188582, A284983, A300284, A392572, A394923, A395597.

%K nonn,cons

%O 1,1

%A _Amiram Eldar_, Apr 30 2026