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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).
1
2, 7, 10, 10, 15, 23, 37, 59, 95, 153, 247, 399, 645, 1043, 1687, 2729, 4415, 7143, 11557, 18699, 30255, 48953, 79207, 128159, 207365, 335523, 542887, 878409, 1421295, 2299703, 3720997, 6020699, 9741695, 15762393, 25504087, 41266479, 66770565, 108037043, 174807607, 282844649
OFFSET
2,1
COMMENTS
Note that F(2*h)/F(h) = Lucas(h) for h > 0. - Editors.
For a guide to related sequences, see A001000. - Clark Kimberling, Aug 07 2012
FORMULA
From Philippe Deléham, Feb 06 2024: (Start)
a(n) = a(n-1) + a(n-2) - 1 for n >= 8.
a(n) = 2*a(n-1) - a(n-3) for n >= 9.
a(n) = 1 + A022112(n-3) for n >= 6.
a(n) = floor(((1 + sqrt(5))/2)*a(n-1)) for n >= 8.
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)).
(End)
MATHEMATICA
leastSeparator[seq_] := Module[{n = 1},
Table[While[Or @@ (Ceiling[n #1[[1]]] <
2 + Floor[n #1[[2]]] &) /@ (Sort[#1, Greater] &) /@
Partition[Take[seq, k], 2, 1], n++]; n, {k, 2, Length[seq]}]];
t = Table[N[Fibonacci[h]/Fibonacci[2 h]], {h, 1, 30}]
t1 = leastSeparator[t]
(* Peter J. C. Moses, Aug 01 2012 *)
CROSSREFS
KEYWORD
nonn
EXTENSIONS
All the terms were corrected by Clark Kimberling, Aug 07 2012
More terms from Sean A. Irvine, Jul 25 2019
STATUS
approved