A097368 Least term in row n of the Fibonacci regression array in A097367.

1, 1, 2, 1, 2, 2, 1, 3, 2, 2, 3, 1, 3, 3, 2, 4, 2, 3, 4, 1, 4, 3, 3, 5, 2, 4, 4, 2, 5, 3, 4, 5, 1, 5, 4, 3, 6, 3, 5, 5, 2, 6, 4, 4, 6, 2, 6, 5, 3, 7, 4, 5, 6, 1, 7, 5, 4, 7, 3, 6, 6, 3, 8, 5, 5, 7, 2, 7, 6, 4, 8, 4, 6, 7, 2, 8, 6, 5, 8, 3, 7, 7, 4, 9, 5, 6, 8, 1, 8, 7, 5, 9, 4, 7, 8, 3, 9, 6, 6, 9, 3, 8, 8, 5, 10, 5

Offset: 2

Author

Clark Kimberling, Aug 09 2004

Keywords

• nonn
• look

Comments

From Robert Israel, Jan 20 2018: (Start)

a(n) = 1 if and only if n is in A000045.

a(n) = 2 if and only if n >= 4 is in A000032 or 2*A000045.

a(m*n) <= m*a(n). (End)

Examples

			Row 8 of the array in A097367 is 7 6 5 4 1 4 6, of which the least term is T(8,5)=1.
		

Links

• Robert Israel, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A000032, A000045, A097367, A097369.

Programs

Maple

T:= proc(n,k)
  local s,t,u;
   s:= n; t:= k;
   do
     u:= s-t;
     if u <= 0 then return t fi;
     s:= t;
     t:= u;
   od;
end proc:
f:= n -> min(seq(T(n,k),k=1..n-1)):
map(f, [$2..200]); # Robert Israel, Jan 19 2018

Mathematica

f[n_] := Fibonacci[n]; d[n_, k_, 1] := n; d[n_, k_, 2] := k;
d[n_, k_, j_] := ((-1)^j) (k*f[j - 1] - n*f[j - 2]);
s[n_, k_] := Select[Range[100], d[n, k, # + 1] <= 0 &, 1];
t = Table[d[n, k, s[n, k]], {n, 2, 20}, {k, 1, n - 1}];  (* A097367 array *)
Flatten[t]  (* A097367 sequence *)
Table[Min[Flatten[Table[d[n, k, s[n, k]], {k, 1, n - 1}]]], {n, 2, 100}]  (* A097368 *)
(* Clark Kimberling, Oct 14 2016 *)

Extensions

a(46) = 6 inserted by Clark Kimberling, Oct 14 2016