A289001 Fixed point of the mapping 00->0010, 01->001, 10->010, starting with 00.

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, 0, 1, 0

Offset

1

Comments

This coincides with the Pell word A171588, the fixed point of the morphism 0->001, 1->0. This was conjectured by R. J. Mathar, Jul 07 2017 and proved by Dekking and Keane in 2022. As observed by Michel Dekking, Mar 09 2022, this also proves the two conjectures about the positions of 0's and 1's stated in the MATHEMATICA section. - N. J. A. Sloane, Mar 09 2022

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

Links

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

Michel Dekking and Mike Keane, Two-block substitutions and morphic words, arXiv:2202.13548 [math.CO], 2022.

Example

The first seven iterates of the mapping:
00
0010
0010010
00100100010
001001000100100010
0010010001001000100100100010010
0010010001001000100100100010010001001001000100100010

Mathematica

z = 10; (* number of iterates *)
s = {0, 0}; w[0] = StringJoin[Map[ToString, s]];
w[n_] := StringReplace[w[n - 1], {"00" -> "0010", "01" -> "001", "10" -> "010"}]
TableForm[Table[w[n], {n, 0, 10}]]
st = ToCharacterCode[w[z]] - 48   (* A289001 *)
Flatten[Position[st, 0]]  (* A001951 conjectured *)
Flatten[Position[st, 1]]  (* A001952 conjectured *)
Table[StringLength[w[n]], {n, 0, 20}] (* A289004 *)

Crossrefs

Cf. A001951, A001952, A289004, A171588.

Sequence in context: A288203 A238470 A286748 * A171588 A289035 A276397

Adjacent sequences: A288998 A288999 A289000 * A289002 A289003 A289004

Keyword

nonn,easy

Author

Clark Kimberling, Jun 25 2017

Status

approved