  Pattern-avoiding ascent sequences have recently been related to set-partition
problems and stack-sorting problems. While the generating functions for several
length-3 pattern-avoiding ascent sequences are known, those avoiding 000, 100,
110, 120 are not known. We have generated extensive series expansions for these
four cases, and analysed them in order to conjecture the asymptotic behaviour.
  We provide polynomial time algorithms for the 000 and 110 cases, and
exponential time algorithms for the 100 and 120 cases. We also describe how the
000 polynomial time algorithm was detected somewhat mechanically given an
exponential time algorithm.
  For 120-avoiding ascent sequences we find that the generating function has
stretched-exponential behaviour and prove that the growth constant is the same
as that for 201-avoiding ascent sequences, which is known.
  The other three generating functions have zero radius of convergence, which
we also prove. For 000-avoiding ascent sequences we give what we believe to be
the exact growth constant. We give the conjectured asymptotic behaviour for all
four cases.
