A261346 Decimal expansion of the side length median of a random triangle of unit inradius.

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

Offset

1,1

Links

Table of n, a(n) for n=1..104.

Steven R. Finch, Three Random Tangents to a Circle, arXiv:1101.3931 (math.PR], 2011.

Formula

Side length density for x>2 is p(x) = (16/Pi^2)*((x*arctan((x + sqrt(x^2 - 4))/2) - x*arctan((x - sqrt(x^2 - 4))/2) + log((x + sqrt(x^2 - 4))/(x - sqrt(x^2 - 4))))/((x^2 + 4)*x)).

Example

5.548203918878445277644299718216988498950141170649469975237323384077...

Mathematica

digits = 104; p[x_] := (16/Pi^2)*((x*ArcTan[(x + Sqrt[x^2 - 4])/2] - x*ArcTan[(x - Sqrt[x^2 - 4])/2] + Log[(x + Sqrt[x^2 - 4])/(x - Sqrt[x^2 - 4])])/((x^2 + 4)*x)); P[x_?NumericQ] := NIntegrate[p[t], {t, 2, x}, WorkingPrecision -> digits + 5]; m = x /. FindRoot[P[x] == 1/2, {x, 5}, WorkingPrecision -> digits + 5]; First[RealDigits[m, 10, digits]]

Crossrefs

Sequence in context: A094245 A117191 A011189 * A011409 A255240 A168578

Adjacent sequences: A261343 A261344 A261345 * A261347 A261348 A261349

Keyword

cons,nonn

Author

Jean-François Alcover, Aug 15 2015

Status

approved