

A088527


Define a Fibonaccitype 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 mth term in some Fibonaccitype 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 mth term in a Fibonaccitype 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}] (* JeanFrançois 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



