A352485 Decimal expansion of the probability that when a unit interval is broken at two points uniformly and independently chosen at random along its length the lengths of the resulting three intervals are the altitudes of a triangle.

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

Offset

0,1

Links

Mohammed Yaseen, Table of n, a(n) for n = 0..10000

Eugen J. Ionascu, Problem 11663, The American Mathematical Monthly, Vol. 119, No. 8 (2012), pp. 699-706; alternative link; Are Random Breaks the Altitudes of a Triangle?, Solution to Problem 116633, by David Farnsworth and James Marento, ibid., Vol. 121, No. 8 (2014), pp. 741-743.

Eugen J. Ionaşcu and Gabriel Prăjitură, Things to do with a broken stick, International Journal of Geometry, Vol. 2, No. 2 (2013), pp. 5-30; arXiv preprint, arXiv:1009.0890 [math.HO], 2010-2013.

Roberto Tauraso, Problem 11663.

Eric Weisstein's World of Mathematics, Altitude.

Formula

Equals 12*sqrt(5)*log((3+sqrt(5))/2)/25 - 4/5.

Equals 24*sqrt(5)*log(phi)/25 - 4/5, where phi is the golden ratio (A001622).

Example

0.23298145831360969333463975908145302101896963809669...

Mathematica

RealDigits[24*Sqrt[5]*Log[GoldenRatio]/25 - 4/5, 10, 100][[1]]

Crossrefs

Cf. A001622, A002390.

Similar sequences: A016627, A019702, A084623, A243398, A339392, A339393, A352484.

Sequence in context: A364895 A299619 A215269 * A356092 A359427 A336246

Adjacent sequences: A352482 A352483 A352484 * A352486 A352487 A352488

Keyword

nonn,cons

Author

Amiram Eldar, Mar 18 2022

Status

approved