A326420 - OEIS

%I #27 Nov 09 2023 19:12:42

%S 1,3,1,3,2,2,1,3,1,3,2,2,1,3,2,1,3,2,1,3,1,3,2,2,1,3,1,3,2,2,1,3,2,1,

%T 3,2,1,3,1,3,2,2,1,3,2,1,3,1,3,2,2,1,3,2,1,3,1,3,2,2,1,3,1,3,2,2,1,3,

%U 2,1,3,2,1,3,1,3,2,2,1,3,1,3,2,2,1,3,2

%N Fixed point of the morphism 1->13, 2->132, 3->1322.

%C The standard form of this sequence, obtained by switching 2 and 3, starts with 1, 2, 1, 2, 3, 3, 1, 2, 1, 2, 3, 3, 1, 2, 3, 1, 2, 3, 1, 2, ...

%C The present version has the property that

%C   a(n) = A285347(n+1) - A285347(n) for n=1,2,....

%C This sequence, as a word, has the remarkable property that it is also fixed point of a uniform morphism of length 3, given by 1->131, 2->132, 3->322.

%C For an algorithm to find this morphism, see Section V of the paper "The spectrum of dynamical systems arising from substitutions of constant length". In this particular case one can verify the truth of this property by noting that the letters 1 and 3 occur in (a(n)) exclusively in the word 13. This implies that one can move the first letter of alpha(3) to the last letter of alpha(1), where alpha is the defining morphism.

%H Paolo Xausa, <a href="/A326420/b326420.txt">Table of n, a(n) for n = 1..10000</a>

%H F. M. Dekking, <a href="http://www.numdam.org/item/?id=PSMIR_1976___2_A6_0">The spectrum of dynamical systems arising from substitutions of constant length</a>, Publications des séminaires de mathématiques et informatique de Rennes, no. 2 (1976), Exposé no. 6, 34 p.

%H F. M. Dekking, <a href="https://doi.org/10.1007/BF00534241">The spectrum of dynamical systems arising from substitutions of constant length</a>, Z. Wahrscheinlichkeitstheorie und verw. Gebiete 41 (1978), 221-239.

%e 1 -> 13 -> 131322 -> 131322131322132132 -> ....

%t Nest[Flatten[ReplaceAll[#,{1->{1,3},2->{1,3,2},3->{1,3,2,2}}]]&,{1},5] (* _Paolo Xausa_, Nov 09 2023 *)

%Y Cf. A285347.

%K nonn

%O 1,2

%A _Michel Dekking_, Sep 12 2019