{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/binary-sequences-meet-the-fibonacci-sequence","title":"Binary sequences meet the Fibonacci sequence","arxiv_id":"2412.11319","date":"2024-12-15","proceeding":null,"authors":["Piotr Miska","Bartosz Sobolewski","Maciej Ulas"],"abstract":"We introduce a new family of meta-Fibonacci sequences $(f(n))_{n\\in\\mathbb{N}}$, governed by the recurrence relation $$f(n)=af(n-u_{n}-1)+bf(n-u_{n}-2),$$ where $\\mathbf{u}=(u_{n})_{n\\in \\mathbb{N}}$ 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 $\\mathcal{V}(f)=\\{h(n): n \\in \\mathbb{N}\\}$ for various $\\mathbf{u}$. We give a sufficient condition for finiteness of $\\mathcal{V}(f)$ and automaticity of $(h(n))_{n \\in \\mathbb{N}}$, which holds in particular when $\\mathbf{u}$ 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 $\\cal{V}(f)$ is infinite for other special binary sequences $\\mathbf{u}$, and obtain a trichotomy in its topological type when $\\mathbf{u}$ is eventually periodic.","url_abs":"https://arxiv.org/abs/2412.11319v2","url_pdf":"https://arxiv.org/pdf/2412.11319v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"binary-sequences-meet-the-fibonacci-sequence","repo_url":"https://github.com/bartoszsobolewski/binary-sequences-meet-fibonacci","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}