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A325172 Up-integers: integers k such that w_(s+1) = ceiling(phi*k) for some k-slow Fibonacci walk, with phi=(1+sqrt(5))/2. See comments for further explanation.

%I A325172 #14 Jun 13 2021 03:25:31
%S A325172 3,4,6,8,11,14,16,17,19,20,21,22,24,25,27,29,30,32,35,37,40,42,45,50,
%T A325172 53,55,58,61,63,66,69,71,74,76,77,79,82,84,85,87,90,92,93,95,97,98,
%U A325172 100,101,103,105,106,108,109,110,111,113,114,116,117,118,119,121,122,124
%N A325172 Up-integers: integers k such that w_(s+1) = ceiling(phi*k) for some k-slow Fibonacci walk, with phi=(1+sqrt(5))/2. See comments for further explanation.
%C A325172 An n-slow Fibonacci walk is a Fibonacci-like sequence that needs a maximum number of steps, s (see A088527), to reach n, and w_(s+1) will be the next term of this sequence. See Chung et al. for further explanation.
%H A325172 Fan Chung, Ron Graham, and Sam Spiro, <a href="https://arxiv.org/abs/1903.08274">Slow Fibonacci Walks</a>, arXiv:1903.08274 [math.NT], 2019. See pp. 3-4.
%o A325172 (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 A325172 dofib(i, j, nb) = {if (nb==2, return (j)); for (k=3, nb, ij = i + j; i = j; j = ij;); return (j);}
%o A325172 s(n) = {my(nb = 2, k); for (i=1, n, for (j=1, n, k = nbs(i, j, n); if (k> nb, nb = k););); nb;} \\ A088527
%o A325172 isup(n) = {my(nb = s(n)); for (i=1, n, for (j=1, n, k = nbs(i, j, n); if (k == nb, w = dofib(i, j, nb+1); if (w == ceil(n*((1+sqrt(5))/2)), return (1));););); return (0);}
%Y A325172 Cf. A001622 (phi), A088527, A325171.
%K A325172 nonn
%O A325172 1,1
%A A325172 _Michel Marcus_, Apr 04 2019