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A024831 a(n) = least m such that if r and s in {F(h)/F(2*h): h = 1,2,...,n} satisfy r < s, then r < k/m < s for some integer k, where F = A000045 (Fibonacci numbers).

%I A024831 #34 Feb 25 2024 13:06:19
%S A024831 2,7,10,10,15,23,37,59,95,153,247,399,645,1043,1687,2729,4415,7143,
%T A024831 11557,18699,30255,48953,79207,128159,207365,335523,542887,878409,
%U A024831 1421295,2299703,3720997,6020699,9741695,15762393,25504087,41266479,66770565,108037043,174807607,282844649
%N A024831 a(n) = least m such that if r and s in {F(h)/F(2*h): h = 1,2,...,n} satisfy r < s, then r < k/m < s for some integer k, where F = A000045 (Fibonacci numbers).
%C A024831 Note that F(2*h)/F(h) = Lucas(h) for h > 0. - Editors.
%C A024831 For a guide to related sequences, see A001000. - _Clark Kimberling_, Aug 07 2012
%H A024831 W. Kuszmaul, <a href="http://arxiv.org/abs/1509.08216">Fast Algorithms for Finding Pattern Avoiders and Counting Pattern Occurrences in Permutations</a>, arXiv preprint arXiv:1509.08216 [cs.DM], 2015-2017.
%F A024831 From _Philippe Deléham_, Feb 06 2024: (Start)
%F A024831 a(n) = a(n-1) + a(n-2) - 1 for n >= 8.
%F A024831 a(n) = 2*a(n-1) - a(n-3) for n >= 9.
%F A024831 a(n) = 1 + A022112(n-3) for n >= 6.
%F A024831 a(n) = floor(((1 + sqrt(5))/2)*a(n-1)) for n >= 8.
%F A024831 G.f.: x^2*(x^6+3*x^5+2*x^4-8*x^3-4*x^2+3*x+2)/((x-1)*(x^2+x-1)).
%F A024831 (End)
%t A024831 leastSeparator[seq_] := Module[{n = 1},
%t A024831 Table[While[Or @@ (Ceiling[n #1[[1]]] <
%t A024831 2 + Floor[n #1[[2]]] &) /@ (Sort[#1, Greater] &) /@
%t A024831 Partition[Take[seq, k], 2, 1], n++]; n, {k, 2, Length[seq]}]];
%t A024831 t = Table[N[Fibonacci[h]/Fibonacci[2 h]], {h, 1, 30}]
%t A024831 t1 = leastSeparator[t]
%t A024831 (* _Peter J. C. Moses_, Aug 01 2012 *)
%Y A024831 Cf. A001000, A001622, A022112.
%K A024831 nonn
%O A024831 2,1
%A A024831 _Clark Kimberling_
%E A024831 All the terms were corrected by _Clark Kimberling_, Aug 07 2012
%E A024831 More terms from _Sean A. Irvine_, Jul 25 2019