{"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/explicit-formulas-for-concatenations-of","title":"Concatenations of Terms of an Arithmetic Progression","arxiv_id":"2201.07127","date":"2022-01-14","proceeding":null,"authors":["Florian Luca","Bertrand Teguia Tabuguia"],"abstract":"Let $(u(n))_{n\\in\\mathbb{N}}$ be an arithmetic progression of natural integers in base $b\\in\\mathbb{N}\\setminus \\{0,1\\}$. We consider the following sequences: $s(n)=\\overline{u(0)u(1)\\cdots u(n) }^b$ formed by concatenating the first $n+1$ terms of $(u(n))_{n\\in\\mathbb{N}}$ in base $b$ from the right; $s_g(n) = \\overline{u(n)u(n-1)\\cdots u(0)}^b$; and $(s_*(n))_{n\\in\\mathbb{N}}$, given by $s_*(0)=u(0)$, $s_*(n)=\\overline{s(n)s_g(n-1)}^b, n\\geq 1$. We construct explicit formulae for these sequences and use basic concepts of linear difference operators to prove they are not P-recursive (holonomic). We also present an alternative proof that follows directly from their definitions. We implemented $(s(n))_{n\\in\\mathbb{N}}$ and $(s_g(n))_{n\\in\\mathbb{N}}$ in the decimal base when $(u(n))_{n\\in\\mathbb{N}}=\\mathbb{N}\\setminus \\{0\\}$.","url_abs":"https://arxiv.org/abs/2201.07127v2","url_pdf":"https://arxiv.org/pdf/2201.07127v2.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":"explicit-formulas-for-concatenations-of","repo_url":"https://github.com/t3gu1a/concatenations","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}