A219642 Number of steps to reach 0 starting with n and using the iterated process: x -> x - (number of 1's in Zeckendorf expansion of x).

0, 1, 2, 3, 3, 4, 4, 5, 6, 6, 7, 7, 7, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 21, 21, 21, 22, 22, 22, 22, 23, 24, 24, 25, 25, 25, 26, 26, 26, 27, 27, 27, 28, 28, 29, 29

Offset

0,3

Comments

See A014417 for the Fibonacci number system representation, also known as Zeckendorf expansion.

Links

A. Karttunen, Table of n, a(n) for n = 0..10946

Formula

a(0)=0; for n>0, a(n) = 1+a(A219641(n)).

Prog

(Scheme with memoization macro definec from Antti Karttunen's Intseq-library):
(definec (A219642 n) (if (zero? n) n (+ 1 (A219642 (A219641 n)))))
(PARI) A007895(n)=if(n<4, n>0, my(k=2, s, t); while(fibonacci(k++)<=n, ); while(k && n, t=fibonacci(k); if(t<=n, n-=t; s++); k--); s)
a(n)=my(s); while(n, n-=A007895(n); s++); s \\ Charles R Greathouse IV, Sep 02 2015
(Python)
from sympy import fibonacci
def a007895(n):
    k=0
    x=0
    while n>0:
        k=0
        while fibonacci(k)<=n: k+=1
        x+=10**(k - 3)
        n-=fibonacci(k - 1)
    return str(x).count("1")
def a219641(n): return n - a007895(n)
l=[0]
for n in range(1, 101):
    l.append(1 + l[a219641(n)])
print(l) # Indranil Ghosh, Jun 09 2017

Crossrefs

Cf. A007895, A014417, A219640, A219641, A219643-A219645, A219648. Analogous sequence for binary system: A071542, for factorial number system: A219652.

Sequence in context: A229790 A156261 A071823 * A139338 A244229 A317596

Adjacent sequences: A219639 A219640 A219641 * A219643 A219644 A219645

Keyword

nonn

Author

Antti Karttunen, Nov 24 2012

Status

approved