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A252230 Triangular array T read by rows: for j = k+1..2*k, k >=1, T(j,k) = least number of iterations of (h,i) -> (i,h-i) needed to take (k,j) to (k',j') satisfying k' <= j'.

%I A252230 #5 Jan 15 2015 11:51:17
%S A252230 1,2,1,2,3,1,2,2,3,1,2,2,4,3,1,2,2,2,3,3,1,2,2,2,4,3,3,1,2,2,2,2,5,3,
%T A252230 3,1,2,2,2,2,4,3,3,3,1,2,2,2,2,2,4,3,3,3,1,2,2,2,2,2,4,5,3,3,3,1,2,2,
%U A252230 2,2,2,2,4,3,3,3,3,1,2,2,2,2,2,2,4,6
%N A252230 Triangular array T read by rows:  for j = k+1..2*k, k >=1, T(j,k) = least number of iterations of (h,i) -> (i,h-i) needed to take (k,j) to (k',j') satisfying k' <= j'.
%C A252230 Max(row n) = A088858(n).  Let F = A000045 (Fibonacci numbers).  Then T(j,k) is the least h such that one of the following holds:  h is odd and F(h+2)/F(h+1) <= j/k, or h is even and F(h+2)/F(h+1) >= j/k.
%H A252230 Clark Kimberling, <a href="/A252230/b252230.txt">Rows n = 1..100, flattened</a>
%e A252230 First 11 rows:
%e A252230 1
%e A252230 2 1
%e A252230 2 3 1
%e A252230 2 2 3 1
%e A252230 2 2 4 3 1
%e A252230 2 2 2 3 3 1
%e A252230 2 2 2 4 3 3 1
%e A252230 2 2 2 2 5 3 3 1
%e A252230 2 2 2 2 4 3 3 3 1
%e A252230 2 2 2 2 2 4 3 3 3 1
%e A252230 2 2 2 2 2 2 4 3 3 3 3 1
%e A252230 Note, for example, that the numbers in row 8 are T(8,9) to T(8,16); e.g., T(8,13) counts these 5 iterations:  (13,8) -> (8,5) -> (5,3) -> (3,2) -> (2,1) -> (1,1).
%t A252230 f[n_] := Fibonacci[n]; h[j_, k_] := Select[Range[40], (OddQ[#] && f[# + 2]/f[# + 1]<= j/k) || (EvenQ[#] && f[# + 2]/f[# + 1] >= j/k) &, 1]; t[k_] := Flatten[Table[h[j, k], {j, k + 1, 2*k}]];
%t A252230 TableForm[Table[t[k], {k, 1, 26}]] ; (* A252230 array *)
%t A252230 Flatten[Table[h[j, k], {k, 1, 100}, {j, k + 1, 2*k}]] (* A252230 sequence *)
%Y A252230 Cf. A000045, A088858
%K A252230 nonn,easy,tabl
%O A252230 1,2
%A A252230 _Clark Kimberling_, Jan 09 2015