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
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