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

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.

Figures (10)

• Figure 1: Values of the sequence $Q(n)$.
• Figure 2: Safety margin $S(n)=n-\max(Q(n-1),Q(n-2))$.
• Figure 3: Difference sequence.
• Figure 4: Dyadic fluctuation profiles shown both separately and in overlay form. For each scale $k$, we plot the normalized points $\left( \frac{m}{\lfloor N/2^k\rfloor}, \;Q(2^k m)-2^{k-1}m \right)$. The individual panels display selected dyadic scales separately, while the overlay panel combines all scales $k=0,\dots,5$ on the same axes.
• Figure 5: Renormalization diagnostic $R(n)=Q(2n)-2Q(n)$. The plot reveals a pronounced parity dependence: the values for odd $n$ remain comparatively small, whereas the values for even $n$ exhibit larger oscillations. This behaviour supports the interpretation of the recursion as a parity-split dyadic renormalization mechanism.

Theorems & Definitions (14)

• Definition 1
• Lemma 1
• proof
• Definition 2
• Definition 3
• Proposition 1
• proof
• Conjecture 1: Dyadic frequency law
• Conjecture 2: Peak location law
• Conjecture 3: Linear Growth