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

1,3

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

Links

Table of n, a(n) for n=1..88.

Formula

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

Example

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;
  ...

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 *)

Prog

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

Crossrefs

Cf. A000045, A003849, A072649, A104326.

Sequence in context: A165079 A175944 A063905 * A295629 A076272 A180101

Adjacent sequences: A130309 A130310 A130311 * A130313 A130314 A130315

Keyword

nonn,easy,tabf

Author

Edwin F. Sampang, May 21 2007

Extensions

More terms from Harvey P. Dale, Apr 18 2012

Status

approved