  In this paper, we study the three-term nested recurrence relation
$B(n)=B(n-B(n-1))+B(n-B(n-2))+B(n-B(n-3))$ subject to initial conditions where
the first $N$ terms are the integers $1$ through $N$. This recurrence is the
three-term analog of Hofstadter's famous $Q$-recurrence
$Q(n)=Q(n-Q(n-1))+Q(n-Q(n-2))$. Nested recurrences are highly sensitive to
their initial conditions. Some initial conditions lead to finite sequences,
others lead to predictable sequences, and yet others lead to sequences that
appear to be chaotic and infinite. A corresponding study to this one was
previously carried out on the $Q$-recurrence. As with that work, we consider
two families of sequences, one where terms with nonpositive indices are
undefined and a second where terms with nonpositive indices are defined to be
zero. We find similar results here as with the $Q$-recurrence, as we can
completely characterize the sequences for sufficiently large $N$. The results
here are, in a sense, simpler, as our sequences are all finite for sufficiently
large $N$.
