  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.
