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URL: https://oeis.org/A394606

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A394606
Decimal expansion of the median of the distribution of distances between two points in a unit disk.
2
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(x) = (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
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
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Mar 26 2026
STATUS
approved