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A285073
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0-limiting word of the morphism 0->10, 1-> 010.
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6
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0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 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, 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, 1, 0, 0, 1, 0, 1
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OFFSET
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1
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COMMENTS
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The morphism 0->10, 1->010 has two limiting words. If the number of iterations is even, the 0-word evolves from 0 -> 10 -> 01010 -> 100101001010 -> 01010100101001010100101001010; if the number of iterations is odd, the 1-word evolves from 0 -> 10 -> 01010 -> 100101001010, as in A285076.
(a(n-1)) is the Sturmian sequence s'(r,1-r) = (ceiling((n+1)r+1-r)-ceiling(nr+1-r))) with r = sqrt(2)-1.
Moreover, a = 01c, where c = A159684 is the Sturmian sequence s(r,r), = (floor((n+2)r)-floor((n+1)r))) fixed point of the morphism gamma: 0 -> 01, 1 -> 010. The morphism gamma is the time-reversal of the morphism psi: 0 -> 10, 1 -> 010 generating (a(n)).
This is a general property: see reference and link. It is also general that the square of psi has fixed point b = 10c, where b = A285076. (End)
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LINKS
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MATHEMATICA
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s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {0, 1, 0}}] &, {0}, 14]; (* A285073 *)
Flatten[Position[s, 0]]; (* A285074 *)
Flatten[Position[s, 1]]; (* A285075 *)
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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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