A394600 - OEIS

A394600

Decimal expansion of the mean distance between two points selected independently at random within the interior of a regular 12-gon with unit circumradius.

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

(list; constant; graph; refs; listen; history; text; internal format)

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 (504 - 660*sqrt(2) + 396*sqrt(3) - 4*sqrt(6) + 2*(3 + sqrt(3)) * sqrt(2 + sqrt(3)) + 6*(1 + 47*sqrt(3)) * sqrt(2 - sqrt(3)) - 27/2 * (15 + 11*sqrt(3)) * log(3) - 4*(27 - sqrt(3)) * log(1 + sqrt(2)) - 4*sqrt(3) * log(2 + sqrt(3)) + 2*(666 + 397*sqrt(3)) * log(2 - sqrt(3)) - (33 - 19*sqrt(3)) * log(2 + sqrt(2 + sqrt(3))) + 3*(899 + 523*sqrt(3)) * log(2 + sqrt(2 - sqrt(3))))/(1080*sqrt(2)).

EXAMPLE

0.884953782114214064113523711258487868874291600958495...

MATHEMATICA

RealDigits[(504 - 660*Sqrt[2] + 396*Sqrt[3] - 4*Sqrt[6] + 2*(3 + Sqrt[3])*Sqrt[2 + Sqrt[3]] + 6*(1 + 47*Sqrt[3])*Sqrt[2 - Sqrt[3]] - 27/2*(15 + 11*Sqrt[3])*Log[3] - 4*(27 - Sqrt[3])*Log[1 + Sqrt[2]] - 4*Sqrt[3]*Log[2 + Sqrt[3]] + 2*(666 + 397*Sqrt[3])*Log[2 - Sqrt[3]] - (33 - 19*Sqrt[3])*Log[2 + Sqrt[2 + Sqrt[3]]] + 3*(899 + 523*Sqrt[3])*Log[2 + Sqrt[2 - Sqrt[3]]])/(1080*Sqrt[2]), 10, 120][[1]]

PROG

(PARI) (504 - 660*sqrt(2) + 396*sqrt(3) - 4*sqrt(6) + 2*(3 + sqrt(3)) * sqrt(2 + sqrt(3)) + 6*(1 + 47*sqrt(3)) * sqrt(2 - sqrt(3)) - 27/2 * (15 + 11*sqrt(3)) * log(3) - 4*(27 - sqrt(3)) * log(1 + sqrt(2)) - 4*sqrt(3) * log(2 + sqrt(3)) + 2*(666 + 397*sqrt(3)) * log(2 - sqrt(3)) - (33 - 19*sqrt(3)) * log(2 + sqrt(2 + sqrt(3))) + 3*(899 + 523*sqrt(3)) * log(2 + sqrt(2 - sqrt(3))))/(1080*sqrt(2))

CROSSREFS

Cf. A091505 (square), A093064 (triangle), A093070 (disk), A394596 (pentagon), A394597 (hexagon), A394598 (octagon), A394599 (10-gon), this constant (12-gon), A394601 (rhombus), A394602 (rectangle).

Sequence in context: A154841 A258984 A112116 * A228719 A021117 A335414

Adjacent sequences: A394597 A394598 A394599 * A394601 A394602 A394603

KEYWORD

nonn,cons

AUTHOR

Amiram Eldar, Mar 26 2026

STATUS

approved