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

Author

N. J. A. Sloane, Nov 20 2003

Keywords

• nonn
• easy
• nice

Comments

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

Links

• 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).

Crossrefs

See A088858 for another version.

Programs

Mathematica

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 *)

PARI

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

Extensions

Corrected and extended by Don Reble, Nov 21 2003