A270788 Unique fixed point of the 3-symbol Fibonacci morphism phi-hat_2.

1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3

Offset

1,2

Comments

Fixed point of the morphism phi-hat_2 given by 1 --> 12, 2 --> 3, 3 --> 12. [Joerg Arndt, Apr 10 2016]

This sequence is the [0->12, 1->3]-transform of the Fibonacci word A003849: if T(0):=12, T(1):=3, then one proves easily with induction that T(phi_1^n(0)) = phi-hat_2^{n+1}(1), and T(phi_1^n(1)) = phi-hat_2^{n+1}(2), where phi_1 denotes the Fibonacci morphism given by 0 --> 01, 1 --> 0. - Michel Dekking, Dec 29 2019

Links

Joerg Arndt, Table of n, a(n) for n = 1..1000

F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.

Shuo Li, Zeckendorf expansion, Dirichlet series and infinite series involving the infinite Fibonacci word, arXiv:2106.05672 [math.NT], 2021.

Formula

Let A(n)=floor(n*tau), B(n)=n+floor(n*tau), i.e., A and B are the lower and upper Wythoff sequences, A=A000201, B=A001950. Then a(n)=1 if n=A(A(k)) for some k; a(n)=2 if n=B(k) for some k; a(n)=3 if n=A(B(k)) for some k. - Michel Dekking, Dec 27 2016

Maple

with(ListTools);
psi:=proc(S)
Flatten(subs( {1=[1, 2], 2=[3], 3=[1, 2]}, S));
end;
S:=[1];
for n from 1 to 10 do S:=psi(S): od:
S;

Mathematica

m = 121; (* number of terms required *)
S[1] = {1};
S[n_] := S[n] = SubstitutionSystem[{1 -> {1, 2}, 2 -> {3}, 3 -> {1, 2}}, S[n-1]];
For[n = 2, True, n++, If[PadRight[S[n], m] == PadRight[S[n-1], m], Print["n = ", n]; Break[]]];
Take[S[n], m] (* Jean-François Alcover, Feb 15 2023 *)

Crossrefs

Cf. A159917 (same sequence if we map 1->2, 2->0, 3->1).

Cf. A000201, A001950, A003849.

Sequence in context: A295561 A076423 A075660 * A190496 A193926 A211450

Adjacent sequences: A270785 A270786 A270787 * A270789 A270790 A270791

Keyword

nonn,easy

Author

N. J. A. Sloane, Mar 30 2016

Extensions

More terms from Joerg Arndt, Apr 10 2016

Offset changed to 1 by Michel Dekking, Dec 27 2016

Status

approved