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.

%I A088527 #15 Apr 04 2019 11:27:08
%S A088527 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,
%T A088527 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,
%U A088527 8,7,8,7,9,8,7,10,8,7,8,7,9,8,7,8,8,9,8,7,11
%N 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.
%C A088527 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?
%C A088527 These sequences are called slow Fibonacci walks by Chung et al. - _Michel Marcus_, Apr 04 2019
%H A088527 Fan Chung, Ron Graham, Sam Spiro, <a href="https://arxiv.org/abs/1903.08274">Slow Fibonacci Walks</a>, arXiv:1903.08274 [math.NT], 2019. See s(n) pp. 1 and 2.
%H A088527 T. Denes, <a href="https://doi.org/10.1016/S0012-365X(03)00207-3">Problem 413</a>, Discrete Math. 272 (2003), 302 (but there are several errors in the table given there).
%t A088527 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 *)
%o A088527 (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);}
%o A088527 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
%Y A088527 See A088858 for another version.
%K A088527 nonn,easy,nice
%O A088527 1,1
%A A088527 _N. J. A. Sloane_, Nov 20 2003
%E A088527 Corrected and extended by _Don Reble_, Nov 21 2003