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

%I A270788 #51 May 22 2025 12:38:53
%S A270788 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,
%T A270788 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,
%U A270788 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
%N A270788 Unique fixed point of the 3-symbol Fibonacci morphism phi-hat_2.
%C A270788 Fixed point of the morphism phi-hat_2 given by 1 --> 12, 2 --> 3, 3 --> 12. [_Joerg Arndt_, Apr 10 2016]
%C A270788 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
%H A270788 Joerg Arndt, <a href="/A270788/b270788.txt">Table of n, a(n) for n = 1..1000</a>
%H A270788 F. Michel Dekking, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Dekking/dekk4.html">Morphisms, Symbolic Sequences, and Their Standard Forms</a>, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
%H A270788 Shuo Li, <a href="https://arxiv.org/abs/2106.05672">Zeckendorf expansion, Dirichlet series and infinite series involving the infinite Fibonacci word</a>, arXiv:2106.05672 [math.NT], 2021.
%F A270788 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
%p A270788 with(ListTools);
%p A270788 psi:=proc(S)
%p A270788 Flatten(subs( {1=[1,2], 2=[3], 3=[1,2]}, S));
%p A270788 end;
%p A270788 S:=[1];
%p A270788 for n from 1 to 10 do S:=psi(S): od:
%p A270788 S;
%t A270788 m = 121; (* number of terms required *)
%t A270788 S[1] = {1};
%t A270788 S[n_] := S[n] = SubstitutionSystem[{1 -> {1, 2}, 2 -> {3}, 3 -> {1, 2}}, S[n-1]];
%t A270788 For[n = 2, True, n++, If[PadRight[S[n], m] == PadRight[S[n-1], m], Print["n = ", n]; Break[]]];
%t A270788 Take[S[n], m] (* _Jean-François Alcover_, Feb 15 2023 *)
%o A270788 (Python)
%o A270788 from math import isqrt
%o A270788 def A270788(n): return (1,3,2)[((m:=(n+2+isqrt(5*(n+2)**2)>>1)-n-2)+isqrt(5*m**2)>>1)-((k:=(n+1+isqrt(5*(n+1)**2)>>1)-n-1)+isqrt(5*k**2)>>1)] # _Chai Wah Wu_, May 22 2025
%Y A270788 Cf. A159917 (same sequence if we map 1->2, 2->0, 3->1).
%Y A270788 Cf. A000201, A001950, A003849.
%K A270788 nonn,easy
%O A270788 1,2
%A A270788 _N. J. A. Sloane_, Mar 30 2016
%E A270788 More terms from _Joerg Arndt_, Apr 10 2016
%E A270788 Offset changed to 1 by _Michel Dekking_, Dec 27 2016