A159917 -id:A159917 - cp's OEIS frontend

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

N. J. A. Sloane, Mar 30 2016

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

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.

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

Cf. A000201, A001950, A003849.

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;

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 *)

from math import isqrt
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

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

More terms from Joerg Arndt, Apr 10 2016

Offset changed to 1 by Michel Dekking, Dec 27 2016

A242082 Nim sequence of game on n counters whose legal moves are removing some number of counters in A027941.

Original entry on oeis.org

0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0

Offset: 0

Nathan Fox, May 03 2014

nonn

Aperiodic, ternary sequence.
Result of applying the map 0->01, 1->2 to A188432.
Let w(1)=01. For all i>1, let w(i)=w(i-1)w(i-1)w(i-2)...w(2)w(1)2 (as a concatenation of words). The limit of this process is this sequence.
Also the Nim sequence of game on n counters whose legal moves are removing either 1 counter or some number of counters in A089910.
a(n+2) = A159917(n), the infinite Fibonacci sequence on {0,1,2}. See also the standard form A270788 of A159917, explaining the formula below. - Michel Dekking, Dec 27 2016

N. Fox, Aperiodic Subtraction Games, Talk given at the Rutgers Experimental Mathematics Seminar, May 01 2014.
U. Larsson, N. Fox, An Aperiodic Subtraction Game of Nim-Dimension Two, Journal of Integer Sequences, 2015, Vol. 18, #15.7.4.

Cf. A027941, A001950, A000201, A026352, A089910.

a(n)=0 if and only if n=0 or n is in A001950.
a(n)=1 if and only if a(n-1)=0, which happens if and only if n is in A026352.
a(n)=2 if and only if n is in A089910.

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

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