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A285966
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{11->1}-transform of the Thue-Morse word A010060.
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3
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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, 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, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0
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OFFSET
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1
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COMMENTS
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This sequence is a morphic sequence, i.e., a letter-to-letter image of a fixed point of a morphism. Coding:
0 followed by 0 by 0,
0 followed by 1 by 2,
1 by 1 if 1 was in 010 in the TM word,
1 by 3 if it was the image of 11 in the TM word.
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
0 -> 21
1 -> 0
2 -> 23
3 -> 210.
Explanation of (the more difficult) first and last productions:
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.
(End)
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LINKS
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EXAMPLE
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As a word, A010060 = 0110100110010110100101100..., and replacing each 11 by 1 gives 01010010010101001010010100100101...
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MATHEMATICA
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s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {0}, 9] (* Thue-Morse, A010060 *)
w = StringJoin[Map[ToString, s]]
w1 = StringReplace[w, {"11" -> "1"}]
st = ToCharacterCode[w1] - 48 (* A285966 *)
Flatten[Position[st, 0]] (* A285967 *)
Flatten[Position[st, 1]] (* A285968 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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