A325171 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.

2, 5, 7, 9, 10, 12, 13, 15, 18, 23, 26, 28, 31, 33, 34, 36, 38, 39, 41, 43, 44, 46, 47, 48, 49, 51, 52, 54, 56, 57, 59, 60, 62, 64, 65, 67, 68, 70, 72, 73, 75, 78, 80, 81, 83, 86, 88, 89, 91, 94, 96, 99, 102, 104, 107, 112, 115, 120, 123, 125, 128, 133, 136, 138, 141, 146, 149

Offset: 1

Author

Michel Marcus, Apr 04 2019

Keywords

• nonn

Comments

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.

Links

• Fan Chung, Ron Graham, and Sam Spiro, Slow Fibonacci Walks, arXiv:1903.08274 [math.NT], 2019. See pp. 3-4.

Crossrefs

Cf. A001622 (phi), A088527, A325172.

Programs

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