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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Inspired by A278586.

LINKS

Table of n, a(n) for n=0..66.

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

Cf. A000045, A002390, A278586.

Sequence in context: A319470 A115001 A338380 * A347647 A008766 A103232

Adjacent sequences:  A309405 A309406 A309407 * A309409 A309410 A309411

KEYWORD

nonn,easy

AUTHOR

A.H.M. Smeets, Jul 29 2019

STATUS

approved

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Last modified September 22 16:50 EDT 2021. Contains 347607 sequences. (Running on oeis4.)