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A024849 a(n) = least m such that if r and s in {|F(h+1)-tau*F(h)|: h = 1,2,...,n} satisfy r < s, then r < k/m < s for some integer k, where F = A000045 (Fibonacci numbers) and tau = (1+sqrt(5))/2 (golden ratio).

%I A024849 #18 Apr 17 2014 03:32:54
%S A024849 2,4,6,9,14,23,36,59,94,153,246,399,644,1043,1686,2729,4414,7143,
%T A024849 11556,18699
%N A024849 a(n) = least m such that if r and s in {|F(h+1)-tau*F(h)|: h = 1,2,...,n} satisfy r < s, then r < k/m < s for some integer k, where F = A000045 (Fibonacci numbers) and tau = (1+sqrt(5))/2 (golden ratio).
%C A024849 For a guide to related sequences, see A001000. - _Clark Kimberling_, Aug 12 2012
%t A024849 f[n_] := Fibonacci[n]; r = GoldenRatio;
%t A024849 leastSeparator[seq_] := Module[{n = 1},
%t A024849 Table[While[Or @@ (Ceiling[n #1[[1]]] <
%t A024849 2 + Floor[n #1[[2]]] &) /@ (Sort[#1, Greater] &) /@
%t A024849 Partition[Take[seq, k], 2, 1], n++]; n, {k, 2, Length[seq]}]];
%t A024849 t = Flatten[Table[Abs[f[h + 1] - r*f[h]], {h, 1, 21}]];
%t A024849 leastSeparator[t]
%t A024849 (* _Peter J. C. Moses_, Aug 01 2012 *)
%Y A024849 Cf. A000045, A001000, A001622.
%K A024849 nonn,more
%O A024849 2,1
%A A024849 _Clark Kimberling_