The Mantovanelli-Hofstadter Sequence

Abstract

We study the perturbed Hofstadter recurrence introduced by Mantovanelli, $\tilde{Q}(n)=\tilde{Q}(n-\tilde{Q}(n-1))+\tilde{Q}(n-\tilde{Q}(n-2))+(-1{)}^n$, with $\tilde{Q}(1)=\tilde{Q}(2)=1$. We prove that this recurrence is well-defined for every positive integer and that ${\lim }_{n\to \infty }\tilde{Q}(n)/n=1/2$. We establish the optimal order $\tilde{Q}(n)=n/2+O(n/\sqrt{\log n})$ and give explicit positive lower and upper bounds for the corresponding normalized limsup. The proof is primarily combinatorial. An odd-even split turns the recurrence into two exact interleavings of binary words, and the resulting Dyck paths define plane forests. Catalan numbers also arise in the enumeration.

arXiv comment50 pages, 3 figures. Minor editorial revisions