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

%I A323116 #26 Nov 10 2023 03:44:13
%S A323116 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,
%T A323116 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,
%U A323116 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
%N A323116 Fixed point of the morphism 1->221, 2->2211.
%C A323116 A self-generating sequence: there are a(n) 1's between successive pairs 22.
%C A323116 (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
%C A323116       beta(11) = 221221
%C A323116       beta(12) = 2212211
%C A323116       beta(21) = 2211221
%C A323116       beta(22) = 22112211.
%C A323116 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).
%C A323116 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.
%H A323116 Robert Israel, <a href="/A323116/b323116.txt">Table of n, a(n) for n = 1..10000</a>
%H A323116 F. M. Dekking, <a href="http://www.jstor.org/stable/44165352">Regularity and irregularity of sequences generated by automata</a>, Séminaire de Théorie des Nombres de Bordeaux (1979-1980), Exp. No. 9, 10 pp., Univ. Bordeaux I, Talence, 1980.
%e A323116 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2,
%e A323116       2,          2,          1,       1,       2,          2,
%p A323116 f(1):= (2,2,1): f(2):= (2,2,1,1):
%p A323116 T:= [2]:
%p A323116 for i from 1 to 5 do T:= map(f,T) od;
%p A323116 T; # _Robert Israel_, Jan 07 2019
%t A323116 Nest[Flatten[ReplaceAll[#,{1->{2,2,1},2->{2,2,1,1}}]]&,{2},4] (* _Paolo Xausa_, Nov 09 2023 *)
%Y A323116 Other self-generating sequences: A000002, A001030, A007538, A006337, A018244, etc.
%K A323116 nonn
%O A323116 1,1
%A A323116 _Michel Dekking_, Jan 05 2019