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From Michel Dekking, Apr 16 2019: (Start)

{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