Connecting Slow Solutions to Nested Recurrences with Linear Recurrent Sequences

TL;DR

This paper generalizes classical sequences related to nested recurrences using linear recurrence relations, connecting combinatorial tree structures, digit representations, and asymptotic analysis.

Contribution

It introduces a unified framework linking nested recurrences, infinite trees, and digit-based frequency sequences for a broad family of linear recurrence parametrized sequences.

Findings

Sequences can be constructed via nested recurrences, trees, or digit strings.

Sequences exhibit specific asymptotic behaviors.

The framework extends classical examples to a larger family.

Abstract

Labeled infinite trees provide combinatorial interpretations for many integer sequences generated by nested recurrence relations. Typically, such sequences are monotone increasing. Several of these sequences also have straightforward descriptions in terms of how often each value in the sequence occurs. In this paper, we generalize the most classical examples to a larger family of sequences parametrized by linear recurrence relations. Each of our sequences can be constructed in three different ways: via a nested recurrence relation, from labeled infinite trees, or by using Zeckendorf-like strings of digits to describe its frequency sequence. We conclude the paper by discussing the asymptotic behaviors of our sequences.