On a perturbed Hofstadter $Q$-recursion

TL;DR

This paper proves that the parity-perturbed Hofstadter Q-sequence is well-defined for all n and exhibits a specific asymptotic behavior, revealing its self-similar structure and potential as a proxy for the original sequence.

Contribution

It establishes the well-definedness and asymptotic properties of the perturbed sequence, linking its structure to Catalan numbers and conjecturing its relation to the original sequence.

Findings

$ ilde{Q}$ is well-defined for all n.

$ ilde{Q}(n)/n$ approaches 1/2 with an error of $O(1/\sqrt{ ext{log} n})$.

Numerical evidence suggests $Q(n) - ilde{Q}(n) = O(n/\sqrt{ ext{log} n})$.

Abstract

The Hofstadter Q-sequence is a prominent example of nested recurrence. Despite decades of study, it is not even known whether Q(n) is defined for all n. Mantovanelli introduced a parity-perturbed variant $\widetilde{Q}$, obtained by adding $(-1)^n$ to the recursion, which surprisingly replaces the chaotic behaviour of Q by an exact dyadic self-similarity. In this paper we prove that $\widetilde{Q}$ is well-defined for all n and satisfies $|\widetilde{Q}(n)/n - 1/2| = O(1/\sqrt{\log n})$. The proof exploits the self-similar structure of the sequence, where alternating arches arise whose frequency combinatorics are governed by the Catalan numbers. A complementary analysis of the arch amplitudes, conditional on two minimal conjectural properties, refines the asymptotic formula to $\limsup_{n\to\infty} |\widetilde{Q}(n)/n - 1/2| \sqrt{\log_2 n} = 1/(3\sqrt{2\pi})$. Numerical experiments suggest the conjecture $Q(n) - \widetilde{Q}(n) = O(n/\sqrt{\log n})$, indicating that $\widetilde{Q}$ may serve as a tractable proxy for Q. This experimental direction will be investigated elsewhere.