%I #39 Nov 23 2024 16:46:35
%S 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,
%T 13,13,13,13,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,
%U 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
%N Each Fibonacci number F(n) appears F(n) times.
%C 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
%C 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
%C a(n) is the largest term in dual Zeckendorf representation of n-1 (A104326), for n >= 2. - _Amiram Eldar_, Aug 09 2024
%F a(n) = A000045(A072649(n)). - _Michel Marcus_, Aug 03 2022
%e As triangle:
%e 1;
%e 1;
%e 2, 2;
%e 3, 3, 3;
%e 5, 5, 5, 5, 5;
%e 8, 8, 8, 8, 8, 8, 8, 8;
%e 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13;
%e ...
%p T:= n-> (f-> f$f)((<<0|1>, <1|1>>^n)[1,2]):
%p seq(T(n), n=1..10); # _Alois P. Heinz_, Nov 23 2024
%t Flatten[Table[#,{#}]&/@Fibonacci[Range[10]]] (* _Harvey P. Dale_, Apr 18 2012 *)
%o (Python)
%o from itertools import islice
%o def A130312_gen(): # generator of terms
%o a, b = 1, 1
%o while True:
%o yield from (a,)*a
%o a, b = b, a+b
%o A130312_list = list(islice(A130312_gen(),20)) # _Chai Wah Wu_, Oct 13 2022
%o (Python)
%o def A130312(n):
%o a, b = 0, 1
%o while (c:=a+b) <= n: a, b = b, c
%o return a # _Chai Wah Wu_, Nov 23 2024
%o (PARI) a(n) = my(m=0); until(fibonacci(m)>n, m++); fibonacci(m-2); \\ _Michel Marcus_, Nov 26 2022
%Y Cf. A000045, A003849, A072649, A104326.
%K nonn,easy,tabf,changed
%O 1,3
%A _Edwin F. Sampang_, May 21 2007
%E More terms from _Harvey P. Dale_, Apr 18 2012