A246105 -id:A246105 - cp's OEIS frontend

A246104 Least m > 0 for which (s(m), ..., s(n+m-1)) = (s(0), ..., s(n)), the first n+1 terms of the infinite Fibonacci word A003849.

2, 3, 5, 5, 8, 8, 8, 13, 13, 13, 13, 13, 21, 21, 21, 21, 21, 21, 21, 21, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89

Offset: 0

Clark Kimberling, Aug 14 2014

If n is a term of A001911, then a(n) = n+2, otherwise a(n) > n+2. - Ivan Neretin, Sep 30 2017

			In A003849, the initial segment (s(0), ..., s(6)) = (0,1,0,0,1,0,1) first repeats at (s(8), ..., s(14)), so that a(6) = 8.

Ivan Neretin, Table of n, a(n) for n = 0..10944

Cf. A000045, A003849, A241422, A246105.

seq(combinat:-fibonacci(n)$combinat:-fibonacci(n-2),n=2..12); # Robert Israel, Oct 01 2017

s = Flatten[Nest[{#, #[[1]]} &, {0, 1}, 12]]; b[m_, n_] := b[m, n] = Take[s, {m, n}]; q = -1 + Flatten[Table[Select[n + Range[2, 1600], b[#, n + # - 1] == b[1, n] &, 1], {n, 1, 120}]]
Flatten@Table[ConstantArray[Fibonacci[n + 1], Fibonacci[n - 1]], {n, 10}] (* Ivan Neretin, Sep 30 2017 *)

Concatenation of F(n - 2) copies of F(n), for n >= 1, where F = A000045 (Fibonacci numbers).

A280514 Index sequence of the reverse block-fractal sequence A003849.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 4, 3, 2, 1, 8, 7, 6, 5, 4, 3, 2, 1, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14

Offset: 1

Clark Kimberling, Jan 06 2017

The sequence is the concatenation of blocks, the n-th of which, for n >=0, consists of the integers from F(n+1) down to F(2) = 1, where F = A000045, the Fibonacci numbers. See A280511 for the definition of reverse block-fractal sequence. The index sequence (a(n)) of a reverse block-fractal sequence (s(n)) is defined (at A280513) by a(n) = least k > 0 such that (s(k), s(k+1), ..., s(k+n)) = (s(n), s(n-1), ..., s(0)).
Apparently (up to offset) a duplicate of A246105. - R. J. Mathar, Jan 10 2017
Let W be the Fibonacci word A003849. Then a(n) is the least k such that the reversal of the first n-block in W occurs in W beginning at the k-th term. Since (a(n)) is unbounded, the reversal of every block in W occurs infinitely many times in W. - Clark Kimberling, Dec 19 2020

			A003849 = (0,1,0,0,1,0,1,0,0,1,0,0,1,...) = (s(1), s(2), ... ).
(init. block #1) = (1); reversal (0) first occurs at s(1), so a(1) = 1;
(init. block #2) = (0,1); rev. (1,0) first occurs at s(2), so a(2) = 2;
(init. block #3) = (0,1,0); rev. (0,1,0) first occurs at s(1), so a(3) = 1;
(init. block #4) = (0,1,0,0); rev. (0,0,1,0) first occurs at s(3), so a(4) = 3.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A000045, A003849.

r = GoldenRatio; t = Table[Floor[(n + 2) #] - Floor[(n + 1) #], {n, 0, 220}] &[
2 - GoldenRatio]  (* A003849 *)
u = StringJoin[Map[ToString, t]]
breverse[seq_] := Flatten[Last[Reap[NestWhile[# + 1 &, 1, (StringLength[
str = StringTake[seq, Min[StringLength[seq], #]]] == # && ! (Sow[StringPosition[seq, StringReverse[str], 1][[1]][[1]]]) === {}) &]]]];
breverse[u]  (* Peter J. C. Moses, Jan 02 2017 *)

cp's OEIS Frontend

A246104 Least m > 0 for which (s(m), ..., s(n+m-1)) = (s(0), ..., s(n)), the first n+1 terms of the infinite Fibonacci word A003849.

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