A289001 - OEIS

%I #37 Mar 09 2022 11:22:23

%S 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,

%T 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,

%U 0,0,1,0,0,0,1,0,0,1,0,0,1,0,0,0,1,0

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

%C 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

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

%H Clark Kimberling, <a href="/A289001/b289001.txt">Table of n, a(n) for n = 1..10000</a>

%H Michel Dekking and Mike Keane, <a href="https://arxiv.org/abs/2202.13548">Two-block substitutions and morphic words</a>, arXiv:2202.13548 [math.CO], 2022.

%e The first seven iterates of the mapping:

%e 00

%e 0010

%e 0010010

%e 00100100010

%e 001001000100100010

%e 0010010001001000100100100010010

%e 0010010001001000100100100010010001001001000100100010

%t z = 10; (* number of iterates *)

%t s = {0, 0}; w[0] = StringJoin[Map[ToString, s]];

%t w[n_] := StringReplace[w[n - 1], {"00" -> "0010", "01" -> "001", "10" -> "010"}]

%t TableForm[Table[w[n], {n, 0, 10}]]

%t st = ToCharacterCode[w[z]] - 48   (* A289001 *)

%t Flatten[Position[st, 0]]  (* A001951 conjectured *)

%t Flatten[Position[st, 1]]  (* A001952 conjectured *)

%t Table[StringLength[w[n]], {n, 0, 20}] (* A289004 *)

%Y Cf. A001951, A001952, A289004, A171588.

%K nonn,easy

%O 1

%A _Clark Kimberling_, Jun 25 2017