Linear Recurrent Subsequences of Meta-Fibonacci Sequences

TL;DR

This paper proves that all linear recurrent sequences with positive coefficients can be embedded within solutions of meta-Fibonacci sequences, answering a previously open question.

Contribution

It establishes that every linear recurrent sequence with positive coefficients can occur as a solution of a meta-Fibonacci sequence, expanding understanding of their relationship.

Findings

Confirmed the existence of meta-Fibonacci solutions for all positive-coefficient linear recurrences.

Extended the theoretical framework connecting linear recurrences and meta-Fibonacci sequences.

Provided a constructive proof for the embedding of linear recurrences in meta-Fibonacci sequences.

Abstract

In a recent paper, Frank Ruskey asked whether every linear recurrent sequence can occur in some solution of a meta-Fibonacci sequence. In this paper, we answer his question in the affirmative for recurrences with positive coefficients.