A394606 - OEIS

%I #7 Apr 14 2026 11:53:32

%S 8,9,1,2,9,0,7,7,5,0,2,5,2,6,7,6,0,1,7,9,5,7,0,6,1,1,6,9,0,7,1,2,2,9,

%T 8,5,1,5,4,5,3,7,5,8,9,3,9,8,8,2,9,5,5,1,3,4,6,5,4,4,9,4,2,8,8,6,9,7,

%U 1,7,9,9,2,8,3,2,4,0,1,2,1,2,8,2,7,0,9,5,1,7,1,1,3,5,9,5,5,8,2,2,1,1,0,7,7

%N Decimal expansion of the median of the distribution of distances between two points in a unit disk.

%C The median of the probability density function f(s) = (1/Pi) * 2*s*(2*theta(s) - sin(2*theta(s))), for 0 <= s <= 2, and 0 otherwise, where theta(s) = arccos(s/2).

%C The unique positive zero of 2*x^2*acos(x/2) + 2*asin(x/2) - x*sqrt(4 - x^2)*(x^2 + 2)/4 - Pi/2.

%H F. Garwood and J. C. Tanner, <a href="https://www.jstor.org/stable/3610457">2800. On Note 2754: A Repeated Integral</a>, The Mathematical Gazette, Vol. 42, No. 342 (1958), pp. 292-293.

%e 0.891290775025267601795706116907122985154537589398829...

%t RealDigits[x /. FindRoot[2*x^2*ArcCos[x/2] + 2*ArcSin[x/2] - x*Sqrt[4 - x^2]*(x^2 + 2)/4 - Pi/2, {x, 1}, WorkingPrecision -> 120]][[1]]

%o (PARI) solve(x = 1/2, 1, 2*x^2*acos(x/2) + 2*asin(x/2) - x*sqrt(4 - x^2)*(x^2 + 2)/4 - Pi/2)

%Y Cf. A093070 (mean), A102519, A394605 (mode), A394608 (square).

%K nonn,cons

%O 0,1

%A _Amiram Eldar_, Mar 26 2026