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
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
KEYWORD
nonn,easy
AUTHOR
Michel Dekking, Sep 09 2022
STATUS
approved