A026608 a(n) = number of 2's between n-th 1 and (n+1)st 1 in A026600.

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

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Comments

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

Examples

			beta(B) = mu(123) = 123231312 = EDC.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..19683
• Jacques Justin and Laurent Vuillon, Return words in Sturmian and episturmian words, RAIRO-Theoretical Informatics and Applications 34.5 (2000): 343-356.

Crossrefs

Cf. A026600, A026608, A026609, A026610, A026611, A026612, A026613.

Programs

Mathematica

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 *)