Nested Recurrence Relations With Conolly-Like Solutions

A nondecreasing sequence of positive integers is \((\alpha,\beta)\)-Conolly, or Conolly-like for short, if for every positive integer \(m\) the number of times that \(m\) occurs in the sequence is \(\alpha + \beta r_m\), where \(r_m\) is \(1\) plus the 2-adic valuation of \(m\). A recurrence relation is \((\alpha, \beta)\)-Conolly if it has an \((\alpha, \beta)\)-Conolly solution sequence. We discover that Conolly-like sequences often appear as solutions to nested (or meta-Fibonacci) recurrence relations of the form \(A(n) = \sum_{i=1}^k A(n-s_i-\sum_{j=1}^{p_i} A(n-a_{ij}))\) with appropriate initial conditions. For any fixed integers \(k\) and \(p_1,p_2,\ldots, p_k\) we prove that there are only finitely many pairs \((\alpha, \beta)\) for which \(A(n)\) can be \((\alpha, \beta)\)-Conolly. For the case where \(\alpha =0\) and \(\beta =1\), we provide a bijective proof using labelled infinite trees to show that, in addition to the original Conolly recurrence, the recurrence \(H(n)=H(n-H(n-2)) + H(n-3-H(n-5))\) also has the Conolly sequence as a solution. When \(k=2\) and \(p_1=p_2\), we construct an example of an \((\alpha,\beta)\)-Conolly recursion for every possible (\(\alpha,\beta)\) pair, thereby providing the first examples of nested recursions with \(p_i>1\) whose solutions are completely understood. Finally, in the case where \(k=2\) and \(p_1=p_2\), we provide an if and only if condition for a given nested recurrence \(A(n)\) to be \((\alpha,0)\)-Conolly by proving a very general ceiling function identity.