A325171 - OEIS

%I #15 Jun 13 2021 03:25:27

%S 2,5,7,9,10,12,13,15,18,23,26,28,31,33,34,36,38,39,41,43,44,46,47,48,

%T 49,51,52,54,56,57,59,60,62,64,65,67,68,70,72,73,75,78,80,81,83,86,88,

%U 89,91,94,96,99,102,104,107,112,115,120,123,125,128,133,136,138,141,146,149

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

%C 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 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 (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 dofib(i, j, nb) = {if (nb==2, return (j)); for (k=3, nb, ij = i + j; i = j; j = ij;); return (j);}

%o 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 isdown(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 == floor(n*((1+sqrt(5))/2)), return (1));););); return (0);}

%Y Cf. A001622 (phi), A088527, A325172.

%K nonn

%O 1,1

%A _Michel Marcus_, Apr 04 2019