A394606 Decimal expansion of the median of the distribution of distances between two points in a unit disk.

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, 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, 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

Offset

0,1

Comments

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).

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.

Links

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

F. Garwood and J. C. Tanner, 2800. On Note 2754: A Repeated Integral, The Mathematical Gazette, Vol. 42, No. 342 (1958), pp. 292-293.

Example

0.891290775025267601795706116907122985154537589398829...

Mathematica

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]]

Prog

(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)

Crossrefs

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

Sequence in context: A376961 A367732 A132718 * A384474 A154904 A155553

Adjacent sequences: A394603 A394604 A394605 * A394607 A394608 A394609

Keyword

nonn,cons

Author

Amiram Eldar, Mar 26 2026

Status

approved