A285966 - OEIS

%I #10 Mar 15 2019 12:28:12

%S 0,1,0,1,0,0,1,0,0,1,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,0,1,0,0,1,0,1,0,0,

%T 1,0,1,0,0,1,0,1,0,1,0,0,1,0,0,1,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,0,1,0,

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

%N {11->1}-transform of the Thue-Morse word A010060.

%C From _Michel Dekking_, Mar 15 2019: (Start)

%C This sequence is a morphic sequence, i.e., a letter-to-letter image of a fixed point of a morphism. Coding:

%C       0 followed by 0 by 0,

%C       0 followed by 1 by 2,

%C       1 by 1 if 1 was in 010 in the TM word,

%C       1 by 3 if it was the image of 11 in the TM word.

%C It then follows that (a(n)) is the letter to letter image 0->0, 1->1, 2->0, 3->1 of the unique fixed point of the morphism

%C       0 -> 21

%C       1 -> 0

%C       2 -> 23

%C       3 -> 210.

%C Explanation of (the more difficult) first and last productions:

%C   010 -> 011001 by the TM morphism, which is coded to 2302. Here the 3 is put in the production 2 -> 23, the 0 is from the production 1 -> 0, and the final 2 is taken care of in either the production 0 -> 21, or the production 2 -> 23. For the last production, note that 0110 -> 01101001 by the TM morphism, which is coded to 232102. Here the prefix 23 is taken care of by the production 2 -> 23, then 11 coded by 3 gives 210, and the final 2 comes from the final 0 in 0110.

%C (End)

%H Clark Kimberling, <a href="/A285966/b285966.txt">Table of n, a(n) for n = 1..10000</a>

%e As a word, A010060 = 0110100110010110100101100..., and replacing each 11 by 1 gives 01010010010101001010010100100101...

%t s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {0}, 9] (* Thue-Morse, A010060 *)

%t w = StringJoin[Map[ToString, s]]

%t w1 = StringReplace[w, {"11" -> "1"}]

%t st = ToCharacterCode[w1] - 48 (* A285966 *)

%t Flatten[Position[st, 0]]  (* A285967 *)

%t Flatten[Position[st, 1]]  (* A285968 *)

%Y Cf. A010060, A285967, A285968.

%K nonn,easy

%O 1

%A _Clark Kimberling_, May 06 2017