A395847 Decimal expansion of the expected distance between the center of a unit-side regular pentagon to a randomly and uniformly selected point on its perimeter.

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

Offset

0,1

Links

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

Index entries for transcendental numbers.

Formula

Equals (sqrt(10*(5 + sqrt(5))) + (5 + 2*sqrt(5))*arccosh(sqrt(5) - 1))/20.

Equals (5+2*sqrt(5))*(tan(Pi/5)*sec(Pi/5) + arcsinh(tan(Pi/5)))/20.

Equals (sqrt(5)*phi^3/20)*(2*tan(Pi/5)/phi + log(2/phi + tan(Pi/5))), where phi is the golden ratio (A001622).

Example

0.744666853936194163107459534588181561974365758435389...

Mathematica

RealDigits[(Sqrt[10*(5 + Sqrt[5])] + (5 + 2*Sqrt[5])*ArcCosh[Sqrt[5] - 1])/20, 10, 120][[1]]

Prog

(PARI) (sqrt(10*(5 + sqrt(5))) + (5 + 2*sqrt(5))*acosh(sqrt(5) - 1))/20

Crossrefs

Cf. A001622, A394596, A395846 (equilateral triangle), A345653 (square), A395848 (regular hexagon).

Sequence in context: A348736 A316161 A377010 * A153349 A395394 A210463

Adjacent sequences: A395844 A395845 A395846 * A395848 A395849 A395850

Keyword

nonn,cons

Author

Amiram Eldar, May 08 2026

Status

approved