This sequence is the fixed point of the morphism 1 -> 12; 2 -> 3; 3 -> "41"; "41" -> 2122. Here, "41" must be treated as a single token. Limits of such morphisms are not usually periodic. The Thue-Morse sequence is a famous example. On Sat, Jul 25, 2020, 4:24 PM Éric Angelini <eric.angelini@skynet.be> wrote:

ONE TWO THREE FOUR ONE TWO ONE TWO TWO THREE ONE TWO THREE THREE FOUR ONE ONE TWO ...

Insert a dot after each vowel:

O.NE. TWO. THRE.E. FO.U.R O.NE. TWO. O.NE. TWO. TWO. THRE.E. O.NE.TWO. THRE.E. THRE.E. FO.U.R O.NE. O.NE. TW.O ...

The size of the successive chunks is given by the sequence itself.

Question: does the said sequence enter into a loop? Best, É.

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