Dyadic Self-Similarity in a Perturbed Hofstadter $Q$-Recursion

TL;DR

This paper investigates a perturbed Hofstadter Q-recursion, revealing unexpected large-scale structure, approximate linear growth, and persistent dyadic self-similarity in the fluctuation term, supported by numerical experiments and heuristic analysis.

Contribution

It uncovers the dyadic self-similarity and large-scale behavior of a perturbed Hofstadter Q-recursion, providing empirical evidence and heuristic explanations for these phenomena.

Findings

Sequence remains well-defined for large n

Sequence approximately grows as n/2

Fluctuation term exhibits dyadic self-similarity

Abstract

We study a perturbed variant of Hofstadter's $Q$-recursion \[ Q(n)=Q(n-Q(n-1))+Q(n-Q(n-2))+(-1)^n, \qquad Q(1)=Q(2)=1 . \] Numerical experiments indicate that the sequence remains well defined for very large values of $n$ and exhibits an unexpectedly structured large-scale behavior. The data provide strong empirical evidence that the sequence grows approximately linearly, with \[ Q(n)\approx \frac{n}{2}. \] Writing $Q(n)=n/2+E(n)$, the fluctuation term $E(n)$ appears to display a persistent dyadic self-similarity: characteristic patterns recur across scales related by powers of two. A heuristic analysis of the recursion suggests a possible explanation for this phenomenon. Since the recursive indices typically lie close to $n/2$, the dynamics repeatedly couple values at scale $n$ with values near scale $n/2$, producing an effective dyadic renormalization mechanism. We further analyze the associated index processes $t_1(n)=n-Q(n-1)$ and $t_2(n)=n-Q(n-2)$, which reveal a pronounced parity dependence in the dynamics. In addition, numerical experiments on the frequency sequence of the values of $Q(n)$ suggest a regular dyadic organization with approximately geometric multiplicities inside blocks $B_k={2^k,\dots,2^{k+1}-1}$. Taken together, these observations point to a possible parity-split dyadic renormalization structure governing the long-term dynamics of the recursion. Establishing rigorous results for these phenomena remains an open problem.