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A280514
Index sequence of the reverse block-fractal sequence A003849.
3
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
1,2
COMMENTS
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
LINKS
EXAMPLE
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.
MATHEMATICA
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 *)
CROSSREFS
Sequence in context: A322263 A279394 A308509 * A246105 A211980 A171730
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jan 06 2017
STATUS
approved