Spot-Based Generations for Meta-Fibonacci Sequences

TL;DR

This paper introduces a novel spot-based generation sequence methodology to systematically identify sequence partitions in meta-Fibonacci sequences, enhancing understanding of their properties including well-behaved and chaotic variants.

Contribution

The paper presents a general, systematic approach for partitioning meta-Fibonacci sequences using spot-based generation sequences, improving upon ad hoc methods.

Findings

Applicable to a broad class of meta-Fibonacci sequences

Effective for both well-behaved and chaotic sequences

Provides new insights into sequence structure and behavior

Abstract

For many meta-Fibonacci sequences it is possible to identify a partition of the sequence into successive intervals (sometimes called blocks) with the property that the sequence behaves "similarly" in each block. This partition provides insights into the sequence properties. To date, for any given sequence, only ad hoc methods have been available to identify this partition. We apply a new concept - the spot-based generation sequence - to derive a general methodology for identifying this partition for a large class of meta-Fibonacci sequences. This class includes the Conolly and Conway sequences and many of their well-behaved variants, and even some highly chaotic sequences, such as Hofstadter's famous Q-sequence.