A130312 Each Fibonacci number F(n) appears F(n) times.

1, 1, 2, 2, 3, 3, 3, 5, 5, 5, 5, 5, 8, 8, 8, 8, 8, 8, 8, 8, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34

Offset: 1

Author

Edwin F. Sampang, May 21 2007

Keywords

• nonn
• easy
• tabf

Comments

Also n-1-s(n-1), where s(n) is the length of the longest proper suffix of p, the length-n prefix of the infinite Fibonacci word (A003849), that appears twice in p. - Jeffrey Shallit, Mar 20 2017

a(n+1) = the least period of the length-n prefix of the infinite Fibonacci word (A003849). A period of a length-n word x is an integer p, 1 <= p <= n such that x[i] = x[i+p] for 1 <= i <= n-p. - Jeffrey Shallit, May 23 2020

a(n) is the largest term in dual Zeckendorf representation of n-1 (A104326), for n >= 2. - Amiram Eldar, Aug 09 2024

Examples

			As triangle:
   1;
   1;
   2,  2;
   3,  3,  3;
   5,  5,  5,  5,  5;
   8,  8,  8,  8,  8,  8,  8,  8;
  13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13;
  ...
		

Crossrefs

Cf. A000045, A003849, A072649, A104326.

Programs

Maple

T:= n-> (f-> f$f)((<<0|1>, <1|1>>^n)[1,2]):
seq(T(n), n=1..10);  # Alois P. Heinz, Nov 23 2024

Mathematica

Flatten[Table[#,{#}]&/@Fibonacci[Range[10]]] (* Harvey P. Dale, Apr 18 2012 *)

PARI

a(n) = my(m=0); until(fibonacci(m)>n, m++); fibonacci(m-2); \\ Michel Marcus, Nov 26 2022

Python

from itertools import islice
def A130312_gen(): # generator of terms
    a, b = 1, 1
    while True:
        yield from (a,)*a
        a, b = b, a+b
A130312_list = list(islice(A130312_gen(),20)) # Chai Wah Wu, Oct 13 2022

Python

def A130312(n):
    a, b = 0, 1
    while (c:=a+b) <= n: a, b = b, c
    return a # Chai Wah Wu, Nov 23 2024

Formula

a(n) = A000045(A072649(n)). - Michel Marcus, Aug 03 2022

Extensions

More terms from Harvey P. Dale, Apr 18 2012