A095098 Fib001 numbers: those k for which the Zeckendorf expansion A014417(k) ends with two zeros and a final one.

6, 9, 14, 19, 22, 27, 30, 35, 40, 43, 48, 53, 56, 61, 64, 69, 74, 77, 82, 85, 90, 95, 98, 103, 108, 111, 116, 119, 124, 129, 132, 137, 142, 145, 150, 153, 158, 163, 166, 171, 174, 179, 184, 187, 192, 197, 200, 205, 208, 213, 218, 221, 226, 229, 234, 239, 242

Offset: 1

Author

Antti Karttunen, Jun 01 2004

Keywords

• nonn
• base

Comments

The asymptotic density of this sequence is sqrt(5)-2. - Amiram Eldar, Mar 21 2022

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A014417, A095086 (fib001 primes).

Set-wise difference of A003622 - A134860.

Programs

Mathematica

a[n_] = 2 Floor[(n + 1) GoldenRatio^2] - n - 3;
a /@ Range[100] (* Jean-François Alcover, Oct 28 2019, after Vladeta Jovovic *)

Python

from sympy import fibonacci
def a(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 x
def ok(n): return str(a(n))[-3:]=="001"
print([n for n in range(1, 501) if ok(n)]) # Indranil Ghosh, Jun 08 2017

Formula

a(n) = 2*floor((n+1)*phi^2)-n-3, where phi = (1+sqrt(5))/2. - Vladeta Jovovic, Jul 05 2004