A394923 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.

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, 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, 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

Offset

0,1

Comments

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

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.

Links

Table of n, a(n) for n=0..104.

Formula

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

Equals A244847 + A002390 * A165954.

Example

0.567748379324468867447924216744999563803471060690635...

Mathematica

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

Prog

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

Crossrefs

Cf. A001622, A002390, A165954, A244847.

Cf. A284983, A394921.

Sequence in context: A385599 A095942 A139395 * A029911 A328941 A205691

Adjacent sequences: A394920 A394921 A394922 * A394924 A394925 A394926

Keyword

nonn,cons

Author

Amiram Eldar, Apr 07 2026

Status

approved