2025/05/30 by Benoit Cloitre, Cloitre, Benoit
Physics and Astronomy · #11B39 #11B83 #Advanced Mathematical Theories and Applications #Combinatorics (math.CO) #FOS: Mathematics #Primary 11B37 #Secondary 05A19
paper · pdf · doi:10.48550/arxiv.2506.00093
openalex publication_date 2025/05/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A family of nested recurrence relations a(n+1) = n - a(m)(n) + a(m+1)(n), parameterized by an integer m ≥ 1 with initial condition a(1)=1, is studied. We prove that a(n)=n-h(n) is the unique solution satisfying this condition, where h(n) is an arithmetical sequence in which each non-negative integer k appears mk+1 times, with h(n) 1-indexed such that h(1)=0. An explicit floor formula for h(n) (and thus for a(n)) is derived. The proof of the main theorem involves establishing a key identity for h(n) that arises from the recurrence; this identity is then proved using arithmetical properties of h(n) and the iterated function a(m)(n) at critical boundary points. Combinatorial interpretations for a(n) and its partial sums (for m=2), and connections to The On-Line Encyclopedia of Integer Sequences (OEIS), including generalizations of Connell's sequence, are also discussed.