A395527 - OEIS

%I #7 Apr 27 2026 09:58:23

%S 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,

%T 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,

%U 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

%N 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.

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

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

%C 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.

%H Norbert Henze, <a href="https://www.jstor.org/stable/3213725">Random Triangles in Convex Regions</a>, Journal of Applied Probability, Vol. 20, No. 1 (1983), pp. 111-125.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SquareTrianglePicking.html">Square Triangle Picking</a>.

%e 0.0571563077778520118877918823626695712206103603711742...

%t 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]]

%o (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

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

%K nonn,cons

%O 0,2

%A _Amiram Eldar_, Apr 27 2026