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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
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
0,4
COMMENTS
Inspired by A278586.
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... .
MATHEMATICA
f[n_] := Length[NestWhileList[# - Ceiling[#/n] &, Fibonacci[n], # > 0 &]] - 1; f /@ Range[0, 70] (* Amiram Eldar, Aug 08 2019 *)
PROG
(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
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
A.H.M. Smeets, Jul 29 2019
STATUS
approved