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
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-François 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
KEYWORD
nonn,easy,nice
AUTHOR
N. J. A. Sloane, Nov 20 2003
EXTENSIONS
Corrected and extended by Don Reble, Nov 21 2003
STATUS
approved