Critical Slow Growth in Averaged Meta-Fibonacci Recursions

TL;DR

This paper studies a family of averaged meta-Fibonacci recursions, revealing regular large-scale behavior, exact structures at critical parameters, and slow growth dynamics, with implications for stability in recursive systems.

Contribution

It introduces a new class of averaged meta-Fibonacci recursions, proves their global well-definedness at critical parameters, and uncovers their asymptotic and structural properties.

Findings

At critical parameter $oldsymbol{oldsymbol{ ext{α=1}}}$, the recursion is well-defined and exhibits a triangular block structure.

The recursion grows approximately as $oldsymbol{oldsymbol{ ext{sqrt(2n)}}}$ asymptotically.

For supercritical $oldsymbol{oldsymbol{ ext{α>1}}}$, linear growth rate must be $oldsymbol{1-rac{1}{α}}$, supported by numerical experiments.

Abstract

We introduce a family of averaged meta-Fibonacci recursions $$ Q_{\alpha,m}(n) = 1+ \left\lfloor \alpha \frac1m \sum_{j=1}^m Q_{\alpha,m}(n-Q_{\alpha,m}(n-j)) \right\rfloor , $$ with initial conditions $$ Q_{\alpha,m}(1)=\cdots=Q_{\alpha,m}(m)=1. $$ Unlike classical Hofstadter-type recursions, the averaging mechanism produces highly regular large-scale behavior. For the critical parameter value $\alpha=1$, we prove global well-definedness for all $m\ge1$, establish an exact triangular block structure, and show that the value $k$ occurs exactly $k$ consecutive times. As a consequence, $$ Q_{1,m}(n)\sim \sqrt{2n}. $$ For the supercritical regime $\alpha>1$, we derive an asymptotic slope constraint showing that any positive linear growth rate, if it exists, must equal $$ 1-\alpha^{-1}. $$ Numerical experiments support the existence of a linear-growth phase and suggest a broader universality phenomenon for generalized averaging operators, including positive-power $L^p$-means. These results indicate that averaging induces a robust regularization mechanism for self-referential recursive systems, leading to stable slow-growth dynamics and nontrivial phase structure.