A309408 Start with X = F(n) = A000045(n). Repeatedly replace X with X - ceiling(X/n); a(n) is the number of steps to reach 0.

0, 1, 1, 2, 3, 5, 7, 10, 13, 18, 23, 29, 36, 44, 53, 63, 74, 85, 98, 111, 125, 140, 157, 174, 192, 211, 231, 251, 273, 296, 319, 343, 369, 395, 423, 451, 481, 510, 541, 573, 606, 640, 675, 710, 747, 785, 823, 863, 903, 944, 987, 1030, 1074, 1119, 1165, 1212, 1260, 1309, 1359, 1409, 1462, 1514, 1568, 1622, 1678, 1734, 1791

Offset: 0

Author

A.H.M. Smeets, Jul 29 2019

Keywords

• nonn
• easy

Comments

Inspired by A278586.

Crossrefs

Cf. A000045, A002390, A278586.

Programs

Mathematica

f[n_] := Length[NestWhileList[# - Ceiling[#/n] &, Fibonacci[n], # > 0 &]] - 1; f /@ Range[0,70] (* Amiram Eldar, Aug 08 2019 *)

PARI

f(x, n) = x - ceil(x/n);
a(n) = my(nb=0, x=fibonacci(n)); while(x, x = f(x, n); nb++); nb; \\ Michel Marcus, Aug 03 2019

Python

n, f1, f0 = 0, 0, 1
while n <= 20000:
    fn, a = f1, 0
    while fn > 0:
        fn, a = fn - (fn+n-1)//n, a+1
    print(n,a)
    n, f1, f0 = n+1, f0, f1+f0

Formula

Lim_{n -> inf} (a(n)/(n^2)) = log(phi) = A002390.

a(n) = n^2*log(phi) - n*log(n) + O(n), phi = (1+sqrt(5))/2.

Lim_{n -> inf} (a(n) - n^2*log(phi) + n*log(n))/ n = -0.4681... .