A394923 - OEIS

%I #5 Apr 07 2026 08:44:33

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

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

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

%N Decimal expansion of the probability that a straight line drawn through a point selected uniformly at random on the perimeter of a regular pentagon, with a direction selected independently and uniformly at random, intersects one of the two sides adjacent to the side on which the point lies.

%C The complementary probability, 0.43225162..., corresponds to an intersection with one of the two opposite sides.

%C The corresponding probability for a square is log(2)/Pi + 1/2 = A284983 + 1/2, and the corresponding probability for a cube is 1 - A394921.

%F Equals (5-sqrt(5))/10 + log(phi) * sqrt(10 + 2*sqrt(5))/(2*Pi), where phi is the golden ratio (A001622).

%F Equals A244847 + A002390 * A165954.

%e 0.567748379324468867447924216744999563803471060690635...

%t RealDigits[(5-Sqrt[5])/10 + Log[GoldenRatio] * Sqrt[10 + 2*Sqrt[5]]/(2*Pi), 10, 120][[1]]

%o (PARI) my(phi = quadgen(5)); (5-sqrt(5))/10 + log(phi) * sqrt(10 + 2*sqrt(5))/(2*Pi)

%Y Cf. A001622, A002390, A165954, A244847.

%Y Cf. A284983, A394921.

%K nonn,cons

%O 0,1

%A _Amiram Eldar_, Apr 07 2026