A323116 Fixed point of the morphism 1->221, 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, 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

1,1

Comments

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.

Links

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.

Example

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,

Maple

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

Mathematica

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

Crossrefs

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

Sequence in context: A014695 A211263 A303827 * A218344 A211272 A298600

Adjacent sequences: A323113 A323114 A323115 * A323117 A323118 A323119

Keyword

nonn

Author

Michel Dekking, Jan 05 2019

Status

approved