On a Family of Nested Recurrences and Their Arithmetical Solutions

TL;DR

This paper investigates a family of nested recurrence relations, providing explicit formulas and combinatorial interpretations, and establishing unique solutions with connections to integer sequences and OEIS.

Contribution

It introduces a novel family of nested recurrences, proves the uniqueness of their solutions, and derives explicit formulas and combinatorial interpretations.

Findings

Explicit floor formula for the sequence h(n) and a(n)

Unique solution characterized by arithmetical properties

Connections to OEIS and generalizations of Connell's sequence

Abstract

A family of nested recurrence relations $a(n+1) = n - a^{(m)}(n) + a^{(m+1)}(n)$, parameterized by an integer $m \ge 1$ 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.