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%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
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