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A287263
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0-limiting word of the morphism 0->11, 1->02, 2->0.
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5
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0, 2, 0, 2, 1, 1, 0, 2, 0, 2, 1, 1, 1, 1, 0, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 2, 0, 2, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 1, 1, 0, 0, 2, 0, 2, 0, 2, 0, 2, 1, 1, 0, 2, 0, 2, 1, 1, 1, 1, 0, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 2, 0, 2, 1, 1
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OFFSET
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1,2
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COMMENTS
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Starting with 0, the first 5 iterations of the morphism yield words shown here:
1st: 11
2nd: 0202
3rd: 110110
4th: 020211020211
5th: 11011002021101100202
The 0-limiting word is the limit of the words for which the number of iterations is even.
Let U, V, W be the limits of u(n)/n, v(n)/n, w(n)/n, respectively. Then 1/U + 1/V + 1/W = 1, where
U = 2.7692923542386314152404094643350334926...,
V = 2.4498438945029551040577327454145475624...,
W = 4.3344900716222708116779374775820643087...
If n >=2, then u(n) - u(n-1) is in {1,2,3,4,6}, v(n) - v(n-1) is in {1,2,5,6,10}, and w(n) - w(n-1) is in {2,4,8,10,16}.
This 0-limiting word and the 1-limiting word A287267 are both fixed points of the irreducible and aperiodic morphism 0->0202, 1->110, 2->11. Therefore they have the same frequencies f0, f1 and f2 of their letters. This implies that the algebraic expressions given for U, V and W in A287267 do also apply to U, V and W above. - Michel Dekking, Oct 09 2019
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LINKS
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EXAMPLE
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2nd iterate: 0202
4th iterate: 020211020211
6th iterate: 020211020211110110020211020211110110
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MATHEMATICA
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s = Nest[Flatten[# /. {0 -> {1, 1}, 1 -> {0, 2}, 2 -> 0}] &, {0}, 10] (* A287263 *)
Flatten[Position[s, 0]] (* A287264 *)
Flatten[Position[s, 1]] (* A287265 *)
Flatten[Position[s, 2]] (* A287266 *)
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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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