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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. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

Table of n, a(n) for n=1..89.

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).

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-Fran├žois Alcover, Jan 15 2013 *)

PROG

(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

CROSSREFS

See A088858 for another version.

Sequence in context: A110007 A327715 A306574 * A030602 A133947 A082090

Adjacent sequences:  A088524 A088525 A088526 * A088528 A088529 A088530

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Nov 20 2003

EXTENSIONS

Corrected and extended by Don Reble, Nov 21 2003

STATUS

approved

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Last modified December 10 23:29 EST 2019. Contains 329910 sequences. (Running on oeis4.)