If you give a "chaotic" math sequence a tiny nudge, it reveals these perfectly repeating patterns hidden inside.
arXiv · March 18, 2026 · 2603.16111
The Takeaway
The Hofstadter Q-sequence is notorious in mathematics for being an unpredictable, messy jumble of numbers. By adding a simple alternating tweak to the formula, researchers discovered that the sequence starts acting like a hall of mirrors, repeating its own behavior across larger and larger scales.
From the 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 dyad