A395527 Decimal expansion of the median of the probability distribution of areas of triangles formed by selecting independently and uniformly three points at random in the interior of a given square of unit area.

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

Offset

0,2

Comments

The mode of the distribution is 0 and its mean is 11/144.

The distribution for a parallelogram is the same as that of a square.

Half the real root of (1/3)*x*(20 - 17*x - 34*x^2*log(x)) - (2/3)*(17*x^2 + 8*x - 1)*(1 - x)*log(1 - x) + 2*x^3*log(x)^2 + 4*x^2 *(3+x)*(PolyLog(2, x) - Pi^2/6) - 1/2 = 0.

Links

Table of n, a(n) for n=0..105.

Norbert Henze, Random Triangles in Convex Regions, Journal of Applied Probability, Vol. 20, No. 1 (1983), pp. 111-125.

Eric Weisstein's World of Mathematics, Square Triangle Picking.

Example

0.0571563077778520118877918823626695712206103603711742...

Mathematica

RealDigits[(x /. FindRoot[(1/3)*x*(20 - 17*x - 34*x^2*Log[x]) - (2/3)*(17*x^2 + 8*x - 1)*(1 - x)*Log[1 - x] + 2*x^3*Log[x]^2 + 4*x^2 *(3 + x)*(PolyLog[2, x] - Pi^2/6) == 1/2, {x, 1/10}, WorkingPrecision -> 120])/2, 10, Automatic, -1][[1]]

Prog

(PARI) solve(x = 1/10, 1/8, (1/3)*x*(20 - 17*x - 34*x^2*log(x)) - (2/3)*(17*x^2 + 8*x - 1)*(1 - x)*log(1 - x) + 2*x^3*log(x)^2 + 4*x^2 *(3+x)*(dilog(x) - Pi^2/6) - 1/2)/2

Crossrefs

Cf. A093072, A093158, A093159, A103281, A103282, A395526.

Sequence in context: A153613 A356706 A318375 * A023571 A160631 A155066

Adjacent sequences: A395524 A395525 A395526 * A395529 A395530 A395531

Keyword

nonn,cons

Author

Amiram Eldar, Apr 27 2026

Status

approved