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).

2, 4, 6, 9, 14, 23, 36, 59, 94, 153, 246, 399, 644, 1043, 1686, 2729, 4414, 7143, 11556, 18699

Offset

2,1

Comments

For a guide to related sequences, see A001000. - Clark Kimberling, Aug 12 2012

Links

Table of n, a(n) for n=2..21.

Formula

Conjectures from Chai Wah Wu, May 15 2026: (Start)

a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4) for n > 7.

G.f.: x^2*(x^5 - x^4 - 3*x^3 - 2*x^2 + 2*x + 2)/((x - 1)*(x + 1)*(x^2 + x - 1)). (End)

Mathematica

f[n_] := Fibonacci[n]; r = GoldenRatio;
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 = Flatten[Table[Abs[f[h + 1] - r*f[h]], {h, 1, 21}]];
leastSeparator[t]
(* Peter J. C. Moses, Aug 01 2012 *)

Crossrefs

Cf. A000045, A001000, A001622.

Sequence in context: A005687 A164139 A218605 * A323227 A283024 A090483

Adjacent sequences: A024846 A024847 A024848 * A024850 A024851 A024852

Keyword

nonn,more

Author

Clark Kimberling

Status

approved