A394599 Decimal expansion of the mean distance between two points selected independently at random within the interior of a regular 10-gon with unit circumradius.

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

Offset

0,1

Links

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

Uwe Bäsel, The moments of the distance between two random points in a regular polygon, arXiv:2101.03815 [math.PR], 2021.

Formula

Equals (4 * (sqrt(5125 + 2110*sqrt(5)) - 24 - 11*sqrt(5)) - (505 + 239*sqrt(5)) * log(2) - 30 * (4 + 3*sqrt(5)) * log(5) - (205 + 107*sqrt(5)) * log(1 + sqrt(5)) + (705 + 343*sqrt(5)) * log(3 + sqrt(5)) - (105 + 47*sqrt(5)) * log(sqrt(5) + sqrt(10 + 2*sqrt(5))) - (65 - 29*sqrt(5)) * log(sqrt(5) + sqrt(10 - 2*sqrt(5))) + (5 + 3*sqrt(5)) * log(5 - sqrt(5) + 2*sqrt(10 - 2*sqrt(5))))/600.

Example

0.876093016045821455437851478188806272714728045761204...

Mathematica

RealDigits[(4*(Sqrt[5125 + 2110*Sqrt[5]] - 24 - 11*Sqrt[5]) - (505 + 239*Sqrt[5])*Log[2] - 30*(4 + 3*Sqrt[5])*Log[5] - (205 + 107*Sqrt[5])*Log[1 + Sqrt[5]] + (705 + 343*Sqrt[5])*Log[3 + Sqrt[5]] - (105 + 47*Sqrt[5])*Log[Sqrt[5] + Sqrt[10 + 2*Sqrt[5]]] - (65 - 29*Sqrt[5])*Log[Sqrt[5] + Sqrt[10 - 2*Sqrt[5]]] + (5 + 3*Sqrt[5])*Log[5 - Sqrt[5] + 2*Sqrt[10 - 2*Sqrt[5]]])/600, 10, 120][[1]]

Prog

(PARI) (4 * (sqrt(5125 + 2110*sqrt(5)) - 24 - 11*sqrt(5)) - (505 + 239*sqrt(5)) * log(2) - 30 * (4 + 3*sqrt(5)) * log(5) - (205 + 107*sqrt(5)) * log(1 + sqrt(5)) + (705 + 343*sqrt(5)) * log(3 + sqrt(5)) - (105 + 47*sqrt(5)) * log(sqrt(5) + sqrt(10 + 2*sqrt(5))) - (65 - 29*sqrt(5)) * log(sqrt(5) + sqrt(10 - 2*sqrt(5))) + (5 + 3*sqrt(5)) * log(5 - sqrt(5) + 2*sqrt(10 - 2*sqrt(5))))/600

Crossrefs

Cf. A091505 (square), A093064 (triangle), A093070 (disk), A394596 (pentagon), A394597 (hexagon), A394598 (octagon), this constant (10-gon), A394600 (12-gon), A394601 (rhombus), A394602 (rectangle).

Sequence in context: A349250 A387303 A215737 * A154718 A334424 A021537

Adjacent sequences: A394596 A394597 A394598 * A394600 A394601 A394602

Keyword

nonn

Author

Amiram Eldar, Mar 26 2026

Status

approved