A280513 -id:A280513 - cp's OEIS frontend

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

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 *)

A280512 Index sequence of the Thue-Morse sequence (A010060, using offset 1) as a reverse block-fractal sequence.

Original entry on oeis.org

1, 3, 2, 1, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 192, 191, 190, 189, 188

Offset: 1

Clark Kimberling, Feb 10 2017

See A280511 for definitions. Note that the records, 3,12,48,192,... are also records in A280510, if A010060 is indexed with offset 1 instead of 0; see Example.

			A010060 = (0,1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0,,...) = (s(1), s(2), ... ).
(init. bl. #1) = (0); reversal = (0), first occurs s(1), so that a(1) = 1;
(init. bl. #2) = (0,1); reversal = (1,0) first occurs at s(3), so that a(2) = 3;
(init. bl. #3) = (0,1,1); reversal = (1,1,0) first occurs s(2), so that a(3) = 2.

Cf. A010060, A280510, A280511, A280513.

seq = Table[Mod[Length[FixedPointList[BitAnd[#, # - 1] &, n]], 2], {n, 0, 400}]  (* A010060 *)
seq = StringJoin[Map[ToString, seq]]
breverse[seq_] := Flatten[Last[Reap[NestWhile[# + 1 &, 1, (StringLength[
str = StringTake[seq, Min[StringLength[seq], #]]] == # && ! (Sow[
StringPosition[seq, StringReverse[str], 1][[1]][[1]]]) === {}) &]]]];
breverse[seq] (* A280512 *) (* Peter J. C. Moses, Jan 01 2017 *)

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

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