A018244 -id:A018244 - cp's OEIS frontend

2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2

Offset: 1

Michel Dekking, Jan 05 2019

nonn

A self-generating sequence: there are a(n) 1's between successive pairs 22.
(a(n)) has some similarity with the Kolakoski sequence A000002. It is the fixed point of a 2-block substitution beta. Beta is simply given by
      beta(11) = 221221
      beta(12) = 2212211
      beta(21) = 2211221
      beta(22) = 22112211.
However, the fact that beta(a) = a is not entirely trivial, as the iterates of beta are ill-defined (since beta^n(12) and beta^n(21) have odd length for all n>0).
By induction one sees that still, beta(beta(...beta(22))) = sigma^n(22), where sigma is the defining morphism given by sigma(1) = 221, sigma(2) = 2211.

			2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2,
      2,          2,          1,       1,       2,          2,

Robert Israel, Table of n, a(n) for n = 1..10000
F. M. Dekking, Regularity and irregularity of sequences generated by automata, Séminaire de Théorie des Nombres de Bordeaux (1979-1980), Exp. No. 9, 10 pp., Univ. Bordeaux I, Talence, 1980.

Other self-generating sequences: A000002, A001030, A007538, A006337, A018244, etc.

f(1):= (2,2,1): f(2):= (2,2,1,1):
T:= [2]:
for i from 1 to 5 do T:= map(f,T) od;
T; # Robert Israel, Jan 07 2019

Nest[Flatten[ReplaceAll[#,{1->{2,2,1},2->{2,2,1,1}}]]&,{2},4] (* Paolo Xausa, Nov 09 2023 *)

cp's OEIS Frontend

A323116 Fixed point of the morphism 1->221, 2->2211.

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