2024/12/15 by Piotr Miska, Miska, Piotr, Bartosz Sobolewski +3
Physics and Astronomy · Mathematics · Computer Science · #Advanced Mathematical Theories and Applications #Advanced Mathematical Theories #Graph Labeling and Dimension Problems
paper · pdf · doi:10.48550/arxiv.2412.11319
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 Walnut. On the other hand, we prove that the set \calV(f) is infinite for other special binary sequences u, and obtain a trichotomy in its topological type when u is eventually periodic.