A243398 -id:A243398 - cp's OEIS frontend

A339392 Numerators of the probability that when a stick is broken up at n-1 points independently and uniformly chosen at random along its length there exist 3 of the n pieces that can form a triangle.

0, 0, 1, 4, 23, 53, 87, 593, 5807, 415267, 8758459, 274431867, 12856077691, 905435186299, 481691519113703, 77763074616922439, 3824113551749834107, 1437016892446437662971, 165559472503434318118655, 146602912901791088694069887, 200050146291129782743679367167

Offset: 1

Amiram Eldar, Dec 04 2020

For the corresponding probability that any triple of pieces can form a triangle, see A001791. The probabilities for these two cases were found by Kong et al. (2013).

			Fractions begin with 0, 0, 1/4, 4/7, 23/28, 53/56, 87/88, 593/594, 5807/5808, 415267/415272, 8758459/8758464, 274431867/274431872, ...
For n = 1 or 2 the number of pieces is less than 3, so the probability is 0.
For n = 3, the stick is being broken into 3 pieces and the probability that they can form a triangle is 1/4, the solution to the classical broken stick problem (see, e.g., Gardner, 2001).

Amiram Eldar, Table of n, a(n) for n = 1..100
Alexander Bogomolny, Stick Broken Into Three Pieces (Trilinear Coordinates), Interactive Mathematics Miscellany and Puzzles, Cut the Knot website.
P. A. Crowdmath, The Broken Stick Project, arXiv:1805.06512 [math.HO], 2018.
Martin Gardner, Probability and Ambiguity, The Colossal Book of Mathematics, W. W. Norton, New York, 2001, chapter 21, pp. 273-285.
Lingyi Kong, Luvsandondov Lkhamsuren, Abigail Turner, Aananya Uppal and A. J. Hildebrand, Random Points, Broken Sticks, and Triangles, Project Report, Illinois Geometry Lab, 2013.

Cf. A000045, A001791, A084623, A234951, A243398, A339393 (denominators).

f = Table[k/(Fibonacci[k + 2] - 1), {k, 2, 20}]; Numerator[1 - FoldList[Times, 1, f]]

a(n) = numerator(1 - Product_{k=2..n} k/(Fibonacci(k+2)-1)).

Lim_{n->oo} a(n)/A339393(n) = 1.

A339393 Denominators of the probability that when a stick is broken up at n-1 points independently and uniformly chosen at random along its length there exist 3 of the n pieces that can form a triangle.

Original entry on oeis.org

1, 1, 4, 7, 28, 56, 88, 594, 5808, 415272, 8758464, 274431872, 12856077696, 905435186304, 481691519113728, 77763074616922464, 3824113551749834112, 1437016892446437662976, 165559472503434318118656, 146602912901791088694069888, 200050146291129782743679367168

Offset: 1

Amiram Eldar, Dec 04 2020

See A339392 for details.

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A000045, A001791, A084623, A234951, A243398, A339392 (numerators).

f = Table[k/(Fibonacci[k + 2] - 1), {k, 2, 20}]; Denominator[1 - FoldList[Times, 1, f]]

a(n) = denominator(1 - Product_{k=2..n} k/(Fibonacci(k+2)-1)).

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.

Original entry on oeis.org

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

Amiram Eldar, Mar 18 2022

			0.23298145831360969333463975908145302101896963809669...

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.

Cf. A001622, A002390.

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

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

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).

cp's OEIS Frontend

A339392 Numerators of the probability that when a stick is broken up at n-1 points independently and uniformly chosen at random along its length there exist 3 of the n pieces that can form a triangle.

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A339393 Denominators of the probability that when a stick is broken up at n-1 points independently and uniformly chosen at random along its length there exist 3 of the n pieces that can form a triangle.

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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.

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Author

Keywords

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Crossrefs

Programs

Mathematica

Formula