Original entry on oeis.org
1, 3, 5, 9, 12, 16
Offset: 1
Original entry on oeis.org
1, 4, 6, 9, 11, 14, 15, 18, 20, 23
Offset: 1
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.
Original entry on oeis.org
0, 0, 1, 4, 23, 53, 87, 593, 5807, 415267, 8758459, 274431867, 12856077691, 905435186299, 481691519113703, 77763074616922439, 3824113551749834107, 1437016892446437662971, 165559472503434318118655, 146602912901791088694069887, 200050146291129782743679367167
Offset: 1
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.
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f = Table[k/(Fibonacci[k + 2] - 1), {k, 2, 20}]; Numerator[1 - FoldList[Times, 1, f]]
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
-
f = Table[k/(Fibonacci[k + 2] - 1), {k, 2, 20}]; Denominator[1 - FoldList[Times, 1, f]]
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