

A026608


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


2



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

1,1


COMMENTS

From Michel Dekking, Apr 16 2019: (Start)
{a(n)} is a morphic sequence, i.e., a lettertoletter 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 3symbol ThueMorse 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


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, RAIROTheoretical Informatics and Applications 34.5 (2000): 343356.


EXAMPLE

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


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 *)


CROSSREFS

Cf. A026600, A026608, A026609, A026610, A026611, A026612, A026613.
Sequence in context: A240592 A001877 A112712 * A264049 A287337 A026612
Adjacent sequences: A026605 A026606 A026607 * A026609 A026610 A026611


KEYWORD

nonn,changed


AUTHOR

Clark Kimberling


STATUS

approved



