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'.

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, 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, 2, 2, 2, 2, 4, 3, 3, 3, 3, 1, 2, 2, 2, 2, 2, 2, 4, 6

Offset

1,2

Comments

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.

Links

Clark Kimberling, Rows n = 1..100, flattened

Example

First 11 rows:
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 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 2 4 3 3 3 3 1
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).

Mathematica

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}]];
TableForm[Table[t[k], {k, 1, 26}]] ; (* A252230 array *)
Flatten[Table[h[j, k], {k, 1, 100}, {j, k + 1, 2*k}]] (* A252230 sequence *)

Crossrefs

Cf. A000045, A088858

Sequence in context: A346913 A269970 A333518 * A036043 A333486 A128628

Adjacent sequences: A252227 A252228 A252229 * A252231 A252232 A252233

Keyword

nonn,easy,tabl

Author

Clark Kimberling, Jan 09 2015

Status

approved