A395213 Decimal expansion of the probability that the unique circle that passes through three points that are independently and uniformly selected at random in the interior of a regular pentagon is entirely contained in that pentagon.

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

Offset

0,1

Comments

The probability that the three points are collinear (i.e., lie on the same straight line) and do not define a circle is 0.

In general, the probability for three points selected in a regular m-gon (m >= 3) is 2*Pi^2/(5*m^2*tan(Pi/m)^2). When m tends to infinity, i.e., when the 3 points are selected in a disk, the probability is 2/5.

Links

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

Fernando Affentranger, Random circles in the d-dimensional unit ball, Journal of Applied Probability , Vol. 26, No. 2 (1989), p. 408-412; JSTOR link.

Index entries for transcendental numbers.

Formula

Equals 2*Pi^2/(125*tan(Pi/5)^2).

Equals 2*Pi^2/(125*(5-2*sqrt(5))).

Example

0.299155951069377931887552491150531223697068484886442...

Mathematica

RealDigits[2*Pi^2/(125*Tan[Pi/5]^2), 10, 120][[1]]

Prog

(PARI) 2*Pi^2/(125*(5-2*sqrt(5)))

Crossrefs

Cf. A019934, A214549 / 10 (equilateral triangle), A091476 / 10 (square), A195055 / 10 (regular hexagon), A395207, A395208, A395209, A395210, A395211, A395212, A395214.

Sequence in context: A011072 A175295 A198141 * A336043 A340723 A201765

Adjacent sequences: A395210 A395211 A395212 * A395214 A395215 A395216

Keyword

nonn,cons

Author

Amiram Eldar, Apr 16 2026

Status

approved