Finding Linear-Recurrent Solutions to Hofstadter-Like Recurrences Using Symbolic Computation

Abstract

The Hofstadter Q-sequence, with its simple definition, has defied all attempts at analyzing its behavior. Defined by a simple nested recurrence and an initial condition, the sequence looks approximately linear, though with a lot of noise. But, nobody even knows whether the sequence is infinite. In the years since Hofstadter published his sequence, various people have found variants with predictable behavior. Oftentimes, the resulting sequence looks complicated but provably grows linearly. Other times, the sequences are eventually linear recurrent. Proofs describing the behaviors of both types of sequence are inductive. In the first case, the inductive hypotheses are fairly ad-hoc, but the proofs in the second case are highly automatable. This suggests that a search for more sequences like these may be fruitful. In this paper, we develop a step-by-step symbolic algorithm to search for these sequences. Using this algorithm, we determine that such sequences come in infinite families that are themselves plentiful. In fact,there are hundreds of easy to describe families based on the Hofstadter Q-recurrence alone.