A395525 Decimal expansion of the median of the distribution of distances between two points in two adjacent unit squares that share a common edge.

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

Offset

1,3

Comments

The real positive root of 3*x^2/2 - 4*x^3/3 + x^4/2 - 3/4 - 2*(1+2*x^2)*sqrt(x^2-1)/3 + 2*x^2*arccos(1/x) = 0.

Links

Table of n, a(n) for n=1..105.

Vangalur S. Alagar, The Distribution of the Distance between Random Points, Journal of Applied Probability, Vol. 13, No. 3 (1976), pp. 558-566. See p. 565, eq. (18).

Example

1.082575483033433462548939194798458707929627054251078...

Mathematica

RealDigits[x /. FindRoot[3*x^2/2 - 4*x^3/3 + x^4/2 - 3/4 - 2*(1+2*x^2)*Sqrt[x^2-1]/3 + 2*x^2*ArcCos[1/x], {x, 1}, WorkingPrecision -> 120]][[1]]

Prog

(PARI) solve(x = 1, 2, 3*x^2/2 - 4*x^3/3 + x^4/2 - 3/4 - 2*(1+2*x^2)*sqrt(x^2-1)/3 + 2*x^2*acos(1/x))

Crossrefs

Cf. A135707 (mean), A395524 (mode), A394608.

Sequence in context: A173158 A379339 A020787 * A179954 A197518 A248300

Adjacent sequences: A395522 A395523 A395524 * A395526 A395527 A395529

Keyword

nonn,cons

Author

Amiram Eldar, Apr 27 2026

Status

approved