cp's OEIS Frontend

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. A proof of this is more involved than the proof for the case of the closely related sequence A026609. The reason is that 2 is not the first letter of A026600.

There are several ways to tackle this. We first remark that it suffices to prove that (a(n)) is the image of a fixed point of a morphism by a morphism delta (instead of a letter-to-letter projection), see Corollary 7.7.5 in the book by Allouche and Shallit.

The sequence A026600 is fixed point of the 3-symbol Thue-Morse morphism mu given by mu: 1->123, 2->231, 3->312. Since the first 2 in (a(n)) is at position 2, we consider the 2-block 3-symbol Thue-Morse morphism mu_2 defined on the set of all nine 2-blocks ij by

1j->12,23,3j, 2j ->23,31,1j , 3j->31,12,2j for j=1,2,3.

We then consider the unique fixed point x = 12,23,32,23,31,13,... of mu_2. The return words of the 'letter' 12 in x are

A:=12,21, B:=12,23,32,23,31,13,31, C:=12,23,32,23,31,11,

D:=12,23,31, E:=12,23,33,31, F:=12,22,23,31,13,31,

G:=12,23,32,23,31, and H:=12,23,31,13,31.

[See Justin & Vuillon (2000) for definition of return word. - N. J. A. Sloane, Sep 23 2019]

The morphism mu_2 induces a morphism beta on the return words given by

A->C, B->BFAEA, C->BFAD, D->BA, E-> BDA, F->GHAEA, G->BFA, H->BAEA.

Counting 3's between 2's in the j's of the ij's in the return words followed by 12 yields the morphism delta given by

delta: A->1, B->02, C->02, D->1, E->1, F->02, G->01, H->2.

Let y = BFAEAGHAEACBD... be the unique fixed point of beta. Then clearly (a(n)) = delta(y).

(End)

The frequencies of 0's, 1's and 2's in (a(n)) are 4/13, 5/13 and 4/13.

This follows from an eigenvector computation, but can also be deduced from the frequency result for the sequence A026609: since the 3-symbol Thue Morse morphism generating sequence A026600 is symmetric under the permutation 1->2->3->1, the two sequences A026609 and A026610 generate the same language, and in particular all subwords have the same frequencies. - Michel Dekking, Apr 16 2019

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