Solutions of certain meta-Fibonacci recurrences

TL;DR

This paper explores solutions to specific meta-Fibonacci recurrences, revealing their connection to counting binary partitions, and provides insights into their behavior based on initial conditions.

Contribution

It introduces new results linking meta-Fibonacci recurrences to binary partition counting, expanding understanding of their solution structures.

Findings

Solutions are closely related to binary partition functions.

The recurrence behavior depends on initial conditions.

New connections between meta-Fibonacci sequences and combinatorial counting.

Abstract

In this note we investigate the solutions of certain meta-Fibonacci recurrences of the form $f(n)=f(n-f(n-1))+f(n-2)$ for various sets of initial conditions. In the case when $f(n)=1$ for $n\leq 1$, we prove that the resulting integer sequence is closely related to the function counting binary partitions of a certain type.