A395391 Decimal expansion of the probability that the segment between two points selected independently and uniformly at random in the interior of a regular pentagram is included entirely within that pentagram.

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

Offset

0,1

Comments

The corresponding probability for a regular hexagram is 23/24 = 0.958333... .

Links

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

Polytope Wiki, Pentagram.

Eric Weisstein's World of Mathematics, Pentagram.

Wikipedia, Pentagram.

Index entries for transcendental numbers.

Formula

Equals (3-phi)/2 + (2-phi)*log(phi), where phi is the golden ratio (A001622).

Equals 1 - 1/(2*phi) + log(phi)/phi^2.

Example

0.874789567009452698532553411765476794061908558500247...

Mathematica

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

Prog

(PARI) my(phi = quadgen(5)); 1 - 1/(2*phi) + log(phi)/phi^2

Crossrefs

Cf. A001622, A002390.

Other constants related to the regular pentagram: A134974, A179050, A188593, A348757.

Sequence in context: A070702 A019870 A388551 * A019903 A327340 A167222

Adjacent sequences: A395388 A395389 A395390 * A395392 A395393 A395394

Keyword

nonn,cons

Author

Amiram Eldar, Apr 21 2026

Status

approved