A394599 - OEIS

%I #5 Mar 26 2026 09:36:14

%S 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,

%T 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,

%U 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

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

%H Uwe Bäsel, <a href="https://arxiv.org/abs/2101.03815">The moments of the distance between two random points in a regular polygon</a>, arXiv:2101.03815 [math.PR], 2021.

%F 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.

%e 0.876093016045821455437851478188806272714728045761204...

%t 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]]

%o (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

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

%K nonn

%O 0,1

%A _Amiram Eldar_, Mar 26 2026