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).

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

Links

Table of n, a(n) for n=2..41.

W. Kuszmaul, Fast Algorithms for Finding Pattern Avoiders and Counting Pattern Occurrences in Permutations, arXiv preprint arXiv:1509.08216 [cs.DM], 2015-2017.

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

Cf. A001000, A001622, A022112.

Sequence in context: A022414 A319932 A236243 * A362861 A194421 A043357

Adjacent sequences: A024828 A024829 A024830 * A024832 A024833 A024834

Keyword

nonn

Author

Clark Kimberling

Extensions

All the terms were corrected by Clark Kimberling, Aug 07 2012

More terms from Sean A. Irvine, Jul 25 2019

Status

approved