A289035 Fixed point of the mapping 00->0010, 01->010, 10->010, starting with 00.

0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0

Offset

1

Comments

Conjecture: the number of letters (0's and 1's) in the n-th iterate of the mapping is given by A289004.

The mapping is applied as in the Mathematica command StringReplace. In particular, for a word of odd length, the final letter is retained, as in 0010010 -> 00100100100. (If the final letter is removed, the iterates are changed, but the limiting fixed point remains unchanged. See A293077.)

More generally, to describe the result of the command StringReplace[w, {u(1)->v(1), u(2)->v(2) ... u(n)-<v(n)}], assume that no u(i) is a subword of any other u(j). The word w is then a concatenation of subwords u(i(0)), g(1), u(i(1)), g(2), u(i(2)), ..., such that u(i(0)) is the empty word, and each g(j) is the subword, g, possibly empty, of least length immediately following u(i(j-1)), such that no subword of g is one of the words u(i). The result of the command is then the concatenation of g(1), v(i(1)), g(2), v(i(2)), ... That is, each u(i) is replaced by v(i), and the words g(i) are left unchanged.

Conjecture: a(n) = A171588(n-3) for n>3. - Michel Dekking, Mar 20 2022

Links

Clark Kimberling, Table of n, a(n) for n = 1..10000

Example

First seven iterates of the mapping:
00
0010
0010010
00100100100
001001001000100100
0010010010001001000100100100010
0010010010001001000100100100010010001001001000100100

Mathematica

z = 10; (* number of iterates *)
s = {0, 0}; w[0] = StringJoin[Map[ToString, s]];
w[n_] := StringReplace[w[n - 1], {"00" -> "0010", "01" -> "010", "10" -> "010"}]
TableForm[Table[w[n], {n, 0, 10}]]
st = ToCharacterCode[w[z]] - 48   (* A289035 *)
Flatten[Position[st, 0]]  (* A289036 *)
Flatten[Position[st, 1]]  (* A289037 *)
Table[StringLength[w[n]], {n, 0, 20}] (* A289004 *)

Crossrefs

Cf. A289036, A289037, A289004, A293076.

Sequence in context: A286748 A289001 A171588 * A276397 A286747 A131531

Adjacent sequences: A289032 A289033 A289034 * A289036 A289037 A289038

Keyword

nonn,easy

Author

Clark Kimberling, Jun 27 2017

Extensions

Updated by Clark Kimberling, Sep 30 2017

Status

approved