A285952 - OEIS

%I #11 Jun 19 2022 04:23:16

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

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

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

%N {0->1, 1->10}-transform of the Thue-Morse word A010060.

%C The morphism {0->1, 1->10} has the infinite Fibonacci word A005614 as fixed point.

%C Since the Thue-Morse word is a concatenation of the 2-blocks 01 and 10, this word is a concatenation of 3-blocks 110 and 101. In particular the frequency of 0 is 1/3. This also follows from the fact that this sequence is a morphic sequence, i.e., the letter-to-letter image of the fixed point of a morphism. The morphism on {1,2,3} can be chosen as 1->123, 2->23, 3->1, and the letter-to-letter map as 1->1, 2->1, 3->0. - _Michel Dekking_, Sep 20 2020

%C Actually this sequence is an automatic sequence. It follows directly from the observations above that (a(n)) is the decoration by the map 0->110, 1->101 of the Thue-Morse word A010060. - _Michel Dekking_, Jun 18 2022

%e As a word, A010060 = 0110100110010110100101100..., and replacing each 0 by 1 and each 1 by 10 gives 110101101110101110110101...

%t s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {0}, 7]  (* Thue-Morse, 010060 *)

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

%t w1 = StringReplace[w, {"0" -> "1", "1" -> "10"}]  (* A285952, word *)

%t st = ToCharacterCode[w1] - 48 (* A285952, sequence *)

%t Flatten[Position[st, 0]]  (* A285953 *)

%t Flatten[Position[st, 1]]  (* A285954 *)

%Y Cf. A010060, A005614, A285950, A285953, A285954.

%K nonn,easy

%O 1

%A _Clark Kimberling_, May 05 2017