Study of Meta-Fibonacci Integer Sequences by Continuous Self-Referential Functional Equations

TL;DR

This paper introduces continuous self-referential functional equations to analyze meta-Fibonacci integer sequences, providing exact models for some and fractal solutions for others, revealing new insights into their complex behaviors.

Contribution

It develops a novel continuous functional equation framework for meta-Fibonacci sequences, including exact solutions and fractal models, advancing understanding of their global behaviors.

Findings

Exact continuous solutions model the backbone of certain sequences.

A fractal solution approach explains the Q-sequence's properties.

The model reproduces anomalous scaling behaviors of the sequences.

Abstract

I propose and investigate the use of continuous functional equations for the study of meta-Fibonacci integer sequences. This exploratory study includes three sequences with quite different behavior: Conway's famous sequence $A(n)= A(A(n-1))+A(n-A(n-1))$, the sequence $D(n)= D(D(n-1))+D(n-1-D(n-2))$ introduced by the present author more than 25 years ago, and Hofstadter's well-known $Q(n)= Q(n-Q(n-1))+Q(n-Q(n-2))$. The sequences are studied in their equivalent detrended forms $(a,d,q)(n)=2\,(A,D,Q)(n)-n$. For $a(n)$ and $d(n)$, a highly symmetric functional equation admits exact continuous solutions that nicely model the global behavior (backbone) of the sequences. For the Hofstadter sequence, a continuous functional model is developed that leads to a random matrix approach for the generation and study of fractal solutions. Two remarkable properties of the Q-sequence are reproduced by the model: the anomalous scaling of the generation length, which scales $\sim (2-\eta)^k$, and the anomalous amplitude growth that scales like $2^{\alpha k}$.