A New Approach to the Hofstadter $Q$-Recurrence

Nested recurrence relations are highly sensitive to their initial conditions. The best-known nested recurrence, the Hofstadter $Q$-recurrence, generates sequences displaying a wide variety of behaviors. Most famous among these is the Hofstadter $Q$-sequence, which appears to be structured at a macro level and chaotic at a micro level. Other choices of initial conditions can lead to more predictable solutions, frequently interleavings of simple sequences. Previous work has focused on the form of a desired solution and on describing an initial condition that generates such a solution. In this paper, we flip this paradigm around. We illustrate how focusing on the form of an initial condition and describing the resulting sequences can yield strange families of new solutions to nested recurrences.