A088858 -id:A088858 - cp's OEIS frontend

A088527 Define a Fibonacci-type sequence to be one of the form s(1) = s_1 >= 1, s(2) = s_2 >= 1, s(n+2) = s(n+1) + s(n); then a(n) = maximal m such that n is the m-th term in some Fibonacci-type sequence.

2, 3, 4, 4, 5, 4, 5, 6, 5, 5, 6, 5, 7, 6, 5, 6, 6, 7, 6, 6, 8, 6, 7, 6, 6, 7, 6, 7, 8, 6, 7, 6, 7, 9, 6, 7, 8, 7, 7, 6, 7, 8, 7, 7, 8, 7, 9, 7, 7, 8, 7, 7, 8, 7, 10, 7, 7, 8, 7, 9, 8, 7, 8, 7, 7, 8, 7, 9, 8, 7, 8, 7, 9, 8, 7, 10, 8, 7, 8, 7, 9, 8, 7, 8, 8, 9, 8, 7, 11

Offset: 1

N. J. A. Sloane, Nov 20 2003

The m-th term in a Fibonacci-type sequence is smallest for the Fibonacci sequence itself. a(Fibonacci(n)) = n (which corresponds to taking s_1 = s_2 = 1). This gives an upper bound a(t) <= log_phi(sqrt(5)*t), roughly. Denes asks: How small can a(n) be and when do small values occur?

These sequences are called slow Fibonacci walks by Chung et al. - Michel Marcus, Apr 04 2019

Fan Chung, Ron Graham, Sam Spiro, Slow Fibonacci Walks, arXiv:1903.08274 [math.NT], 2019. See s(n) pp. 1 and 2.
T. Denes, Problem 413, Discrete Math. 272 (2003), 302 (but there are several errors in the table given there).

See A088858 for another version.

max = 12; s[n_] := (1/2)*((3*s1 - s2)*Fibonacci[n] + (s2 - s1)*LucasL[n]); a[n_] := Reap[ Do[If[s[m] == n, Sow[m]], {m, 1, max}, {s1, 1, max}, {s2, 1, max}]][[2, 1]] // Max; Table[a[n], {n, 1, 89}] (* Jean-François Alcover, Jan 15 2013 *)

nbs(i, j, n) = {my(nb = 2, ij); until (j >= n, ij = i+j; i = j; j = ij; nb++); if (j==n, nb, -oo);}
a(n) = {my(nb = 2, k); for (i=1, n, for (j=1, n, k = nbs(i, j, n); if (k> nb, nb = k););); nb;} \\ Michel Marcus, Apr 04 2019

Corrected and extended by Don Reble, Nov 21 2003

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'.

Original entry on oeis.org

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

Clark Kimberling, Jan 09 2015

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.

			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).

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

Cf. A000045, A088858

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 *)

cp's OEIS Frontend

A088527 Define a Fibonacci-type sequence to be one of the form s(1) = s_1 >= 1, s(2) = s_2 >= 1, s(n+2) = s(n+1) + s(n); then a(n) = maximal m such that n is the m-th term in some Fibonacci-type sequence.

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