A356982 Fixed point of the morphism 0->010, 1->000.

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

Offset

1

Comments

An element of the list of "type 3-3" sequences (see A189628).

(a(n)) has the same set of subsequences as A182581.

Proof: A182581 is fixed point of sigma: 0->001, 1->000 . Let tau: 0->010, 1->000. Then sigma and tau are conjugate morphisms:

for any 0-1-word w one has sigma(w) 0 = 0 tau(w). With induction one proves from this that for all n>1

P(n) tau^{n+1}(0) = sigma^{n+1}(0) P(n),

P(n-1) tau^{n+1}(1) = sigma^{n+1}(1) P(n-1),

where P(n) := sigma^{n}(0)sigma^{n-1}(0)...sigma(0)0.

This implies that the fixed points have the same subwords.

Links

Paolo Xausa, Table of n, a(n) for n = 1..10000

Mazen Khodier, New Methods for Analyzing the Properties of Automatic Sequences, Master's Thesis, Univ. Waterloo (Canada 2026). See p. 27, Table 5.1.

Formula

a(n) = A253786(n) mod 2. - Kevin Ryde, Sep 09 2022

Mathematica

With[{nmax=200}, Mod[IntegerExponent[2Range[nmax]-1, 3], 2]] (* or *)
Nest[Flatten[ReplaceAll[#, {0->{0, 1, 0}, 1->{0, 0, 0}}]]&, {0}, 5] (* Paolo Xausa, Nov 09 2023 *)

Prog

(PARI) a(n) = n--; my(ret=0, r); while([n, r]=divrem(n, 3); r==1, ret=!ret); ret; \\ Kevin Ryde, Sep 13 2022

Crossrefs

Cf. A189628, A182581, A253786.

Sequence in context: A355202 A353556 A228495 * A284683 A369968 A288673

Adjacent sequences: A356979 A356980 A356981 * A356983 A356984 A356985

Keyword

nonn,easy

Author

Michel Dekking, Sep 09 2022

Status

approved