Search: a026612 -id:a026612
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A026608
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a(n) = number of 2's between n-th 1 and (n+1)st 1 in A026600.
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+10
2
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2, 0, 2, 0, 1, 1, 1, 2, 1, 0, 1, 1, 1, 2, 0, 2, 0, 1, 1, 2, 0, 2, 0, 2, 0, 1, 2, 0, 1, 1, 1, 2, 0, 2, 0, 1, 1, 2, 0, 2, 0, 2, 0, 1, 1, 2, 0, 2, 0, 1, 1, 1, 2, 0, 1, 2, 0, 2, 0, 2, 0, 1, 1, 2, 0, 2, 0, 1, 1, 1, 2, 1, 0, 1, 1, 1, 2, 0, 2, 0, 2, 0, 1, 1, 1, 2, 0, 2, 0, 1
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OFFSET
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1,1
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COMMENTS
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{a(n)} is a morphic sequence, i.e., a letter-to-letter projection of a fixed point of a morphism. This follows from a study of the return words of 1 in {a(n)}: the word 1 in {a(n)} has 7 return words. These are A:=1, B:=123, C:=12, D:=13, E:=12323, F:=1233, and G:=1223.
[See Justin & Vuillon (2000) for definition of return word. - N. J. A. Sloane, Sep 23 2019]
The sequence A026600 is a fixed point of the 3-symbol Thue-Morse morphism mu given by mu: 1->123, 2->231, 3->312.
This induces a morphism beta on the return words given by
beta: A->B, B->EDC, C->EA, D->FC, E->EDGDC, F->EDBC, G->EBDC.
Counting 2's in the return words yields the morphism gamma given by
gamma: A->0, B->1, C->1, D->0, E->2, F->1, G->2.
Let y = EDGDCFCEBDCf... be the unique fixed point of beta. Then clearly (a(n)) = gamma(y).
(End)
The frequencies of 0's, 1's and 2's in {a(n)} are 4/13, 5/13 and 4/13, despite the fact that the gamma above is different from the gamma in A026609. However, the languages of the words A026609 and {a(n)} are different. The word 20201 does appear in A026608, A026611, and A026612, but not in the other triple of sequences A026609, A026610 and A026613. - Michel Dekking, Apr 16 2019
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LINKS
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EXAMPLE
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beta(B) = mu(123) = 123231312 = EDC.
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MATHEMATICA
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Map[Count[#, 2] &, DeleteCases[SplitBy[#, # == 1 &], _?(# == {1} &)]] &@ Nest[Flatten[# /. {1 -> {1, 2, 3}, 2 -> {2, 3, 1}, 3 -> {3, 1, 2}}] &, {1}, 6] (* Michael De Vlieger, Apr 16 2019, after Robert G. Wilson v at A026600 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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