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A288173
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Fixed point of the mapping 00->0010, 1->001, starting with 00.
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4
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0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1
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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., the letter-to-letter image of a fixed point of a morphism mu. In fact, one can take the alphabet {1,2,...,8}, with the morphism
mu: 1->123, 2->14, 3->56, 4->1, 5->27, 6->83, 7->56, 8->14,
and the letter-to-letter map lambda defined by
lambda: 1->0, 2->0, 3->1, 4->0, 5->0, 6->1, 7->0, 8->0.
Then (a(n)) = lambda(x), where x = 12314561231278... is the unique fixed point of the morphism mu.
How does one see this? We note first that the mapping SR: 00->0010, 1->010, is an algorithmic procedure given by StringReplace in Mathematica. This makes it hard in general to describe iterates of it.
To circumvent this, we do the following: define the three words
A:=001, B:=0001, C:=00001.
Then (a(n)) is a concatenation of these three words, and moreover the StringReplace algorithm SR acts context free on concatenations of these three words:
SR(001) = 0010001, SR(0001) = 00100001, SR(00001) = 00100010001.
This induces a morphism alpha on the alphabet {A,B,C} given by
alpha(A) = AB, alpha(B) = AC, alpha(C) = ABB.
It is clear that a(n) = delta(t(n)), where delta is given by:
delta(A) = 001, delta(B) = 0001, delta(C) = 00001,
and t is the unique fixed point t = ABACABABB.... of the morphism tau.
In my paper "Morphic words, Beatty sequences and integer images of the Fibonacci language" such a map delta is called a decoration. It is well-known that decorated fixed points are morphic sequences, and the 'natural' algorithm to achieve this yields a morphism on an alphabet of 3+4+5 = 12 symbols. The algorithm gives a large freedom in defining the morphism. This permits to reduce the size of the alphabet, and to obtain the morphism mu and the letter-to-letter map lambda given above.
See A288176 for a proof of Kimberling's conjecture on the number of letters (0's and 1's) in the n-th iteration of the mapping SR. (End)
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LINKS
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EXAMPLE
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Iterates, starting with 00:
00
0010
00100010
0010001001000010
0010001001000010010001001000100010
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MATHEMATICA
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s = {0, 0}; w[0] = StringJoin[Map[ToString, s]];
w[n_] := StringReplace[w[n - 1], {"00" -> "0010", "1" -> "001"}]
Table[w[n], {n, 0, 8}]
st = ToCharacterCode[w[11]] - 48 (* A288173 *)
Flatten[Position[st, 0]] (* A288174 *)
Flatten[Position[st, 1]] (* A288175 *)
Table[StringLength[w[n]], {n, 1, 35}] (* A288176 conjectured *)
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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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