A189640 Fixed point of the morphism 0->001, 1->101.

0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0

Offset

1

Comments

Are the positions of 0 given by A026138? Are the positions of 1 given by A026232?

This sequence appears to be: 0, followed by A080846. [R. J. Mathar, May 16 2011]

From Michel Dekking, Oct 26 2019: (Start)

Proof of Mathar's third conjecture: Let alpha be the defining morphism for (a(n)): alpha(0)=001, alpha(1)=101. Let beta be the defining morphism for A080846: beta(0)=010, beta(1)=011.

To prove the conjecture, it suffices to prove that

alpha^n(0) 0 = 0 beta^n(0) for all n>0.

This holds for n=1. The important property of the morphisms alpha and beta is that

10^{-1}alpha^n(0) = alpha^n(1) and beta^n(0) = beta^n(1) 1^{-1}0 for all n>0.

Here 0^{-1} and 1^{-1} are the free group inverses of 0 and 1.

We use this in the induction step:

alpha^{n+1}(0) 0 = alpha^n(00)alpha^n(1) 0 =

alpha^n(0) alpha^n(0) 10^{-1}alpha^n(0) 0 =

0 beta^n(0) 0^{-1} 0 beta^n(0) 0^{-1} 10^{-1} 0 beta^n(0) 0^{-1} 0 =

0 beta^n(0) beta^n(1)1^{-1}0 0^{-1} 1 beta^n(0) =

0 beta^n(010) = 0 beta^{n+1}(0).

(End)

From Michel Dekking, Oct 26 2019: (Start)

Let beta be the defining morphism for this sequence. It is notationally convenient to write the morphism beta on the alphabet {A,B}:

beta: A -> AAB, B -> BAB.

Let delta be the 'decoration' morphism defined by delta(A) = 0, delta(B) = 10.

Let x = AABAABBABA... be the fixed point of beta.

CLAIM: delta(x) = A049320, the non-primitive Chacon sequence.

This is proved in Ferenczi, 1995. Here in an independent, short proof.

Let gamma given by 0->0010, 1->1 be the Chacon morphism, and let

c = 0010001010010... be the fixed point of gamma. One verifies that

delta(AAB) = delta(AABAABBAB) = gamma(0010),

delta(BAB) = delta(BABAABBAB) = gamma(10010).

So delta(beta(A)) = gamma(delta(A)), and delta(beta(B)) = gamma(delta(B)).

It follows that

delta(beta^n(A)) = gamma^n(delta(A)) = gamma^n(0) for all n>0.

The left side converges to the delta image of the fixed point x of beta, the right side to c, the non-primitive Chacon sequence.

We have shown that the [10->1]-transform of A049320 equals (a(n)).

(End)

A generalized choral sequence c(3n+r_0)=0, c(3n+r_1)=1, c(3n+r_c)=c(n), with r_0=1, r_1=2, r_c=0, and c(0)=0. - Joel Reyes Noche, Jun 14 2021

References

Joel Reyes Noche, Generalized Choral Sequences, Matimyas Matematika, 31(2008), 25-28.

Links

Table of n, a(n) for n=1..122.

S. Ferenczi, Les transformations de Chacon: combinatoire, structure géométrique, lien avec les systèmes de complexité 2n + 1, Bull. Soc. Math. France 123 (1995), 271-292.

Joel Reyes Noche, On generalized choral sequences, Gibon, IX(2011), 51-69.

Formula

a(3k-2)=a(k), a(3k-1)=0, a(3k)=1 for k>=1, a(0)=0.

Example

Start: 0
Rules:
  0 --> 001
  1 --> 101
-------------
0:   (#=1)
  0
1:   (#=3)
  001
2:   (#=9)
  001001101
3:   (#=27)
  001001101001001101101001101
4:   (#=81)
  001001101001001101101001101001001101001001101101001101101001101001001101101001101

Mathematica

t = Nest[Flatten[# /. {0->{0, 0, 1}, 1->{1, 0, 1}}] &, {0}, 5] (*A189640*)
f[n_] := t[[n]]
Flatten[Position[t, 0]] (*A026138*)
Flatten[Position[t, 1]] (*A026232*)
s[n_] := Sum[f[i], {i, 1, n}]; s[0] = 0;
Table[s[n], {n, 1, 120}] (*A189641*)

Prog

(PARI) a(n) = my(r); if(n--, until(r, [n, r]=divrem(n, 3))); r==2; \\ Kevin Ryde, Nov 09 2021

Crossrefs

Cf. A080846 (essentially the same), A026138, A026232, A189641.

Cf. A189628 (guide).

Sequence in context: A286804 A188071 A343219 * A289057 A106138 A273129

Adjacent sequences: A189637 A189638 A189639 * A189641 A189642 A189643

Keyword

nonn,easy

Author

Clark Kimberling, Apr 24 2011

Status

approved