Papers › Binary sequences meet the Fibonacci sequence
Binary sequences meet the Fibonacci sequence
Piotr Miska, Bartosz Sobolewski, Maciej Ulas
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We introduce a new family of meta-Fibonacci sequences (f(n))_(n∈ℕ), governed by the recurrence relation f(n)=af(n-uₙ-1)+bf(n-uₙ-2), where 𝐮=(uₙ)_(n∈ℕ) is a sequence with values $0,1$. Our study focuses on the properties of the sequence of quotients h(n) = f(n+1)/f(n) and its set of values 𝒱(f)={h(n): n ∈ℕ} for various 𝐮. We give a sufficient condition for finiteness of 𝒱(f) and automaticity of (h(n))_(n ∈ℕ), which holds in particular when 𝐮 is the famous Prouhet-Thue-Morse sequence. In the automatic case, a constructive approach is used, with the help of the software \texttt{Walnut}. On the other hand, we prove that the set V(f) is infinite for other special binary sequences 𝐮, and obtain a trichotomy in its topological type when 𝐮 is eventually periodic.
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