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A285341 Fixed point of the morphism 0 -> 10, 1 -> 1011. 5
1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1
COMMENTS
From _Michel Dekking, Sep 08 2019: (Start)
A short proof of Mathar's conjecture can be obtained by using the fact that sigma and tau are conjugate morphisms. For all words w one has
tau(w) = 1^{-1} sigma(w) 1.
The proof is again by induction. Suppose that
(A) : 0 sigma^n(0) = tau^n(0) 0
holds. Then, choosing w = tau^n(0):
tau^{n+1}(0) 0 = tau(tau^n(0)) 0 =
1^{-1} sigma(tau^n(0)) 1 0 =
1^{-1} sigma(0 sigma^n(0) 0^{-1}) 10 =
1^{-1} 10 sigma^{n+1}(0) 0^{-1}1^{-1} 10 =
0 sigma^{n+1}.
So (A) holds for n+1.
(End)
LINKS
FORMULA
Conjecture: a(n) = A284893(n+1). - R. J. Mathar, May 08 2017
From Michel Dekking, Feb 05 2018: (Start)
Proof of this conjecture: let sigma be the morphism 0 -> 10, 1 -> 1011, and let tau be the morphism 0 -> 01, 1 -> 0111, which has A284893 as a fixed point. It clearly suffices to prove the relation, for all n=1,2,3,...:
(A) : 0 sigma^n(0) = tau^n(0) 0
To prove such a thing one needs a second relation, for all n=1,2,3,...:
(B) : 0 sigma^n(1) 0^{-1} = tau^n(0) tau^n(1) [tau^n(0)]^{-1}
Here 0^{-1} and [tau^n(0)]^{-1} are the free group inverses of 0 and tau^n(0).
For n=1, we do indeed have:
(a) 0 sigma(10) = 0101110 = 0101110 = tau(01)0
(b) 0 sigma(1) 0^{-1} tau(0) = 010111 = tau(0)tau(1).
Using the induction hypothesis with (A) and (B) in the second line, one obtains
0 sigma^{n+1}(0) = 0 sigma^n(1) 0^{-1} 0 sigma^n(0)
= tau^n(0) tau^n(1) [tau^n(0)]^{-1} tau^n(0) 0
= tau^{n+1}(0) 0.
Similarly,
0 sigma^{n+1}(1) 0^{-1}
= 0 sigma^n(1) 0^{-1} 0 sigma^n(0) 0^{-1} 0 sigma^n(11) 0^{-1}
= tau^n(0)tau^n(1)[tau^n(0)]^{-1} tau^n(0) 0 0^{-1}
tau^n(0) tau^n(11) [tau^n(0)]^{-1}
= tau^{n+1}(0) tau^{n+1}(1) [tau^n(1)]^{-1}[tau^n(0)]^{-1}
= tau^{n+1}(0) tau^{n+1}(1) [tau^{n+1}(0)]^{-1}. (End)
EXAMPLE
0 -> 10-> 1011 -> 10111010111011.
MATHEMATICA
s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {1, 0, 1, 1}}] &, {0}, 10]; (* A285341 *)
u = Flatten[Position[s, 0]]; (* A285342 *)
Flatten[Position[s, 1]]; (* A285343 *)
u/2 (* A285344)
CROSSREFS
Sequence in context: A089045 A259599 A070749 * A059778 A355690 A359590
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 25 2017
STATUS
approved

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Last modified July 21 11:45 EDT 2024. Contains 374472 sequences. (Running on oeis4.)