  We introduce an algorithmic approach based on generating tree method for
enumerating the inversion sequences with various pattern-avoidance
restrictions. For a given set of patterns, we propose an algorithm that outputs
either an accurate description of the succession rules of the corresponding
generating tree or an ansatz. By using this approach, we determine the
generating trees for the pattern-classes $I_n(000, 021), I_n(100, 021)$,
$I_n(110, 021), I_n(102, 021)$, $I_n(100,012)$, $I_n(011,201)$, $I_n(011,210)$
and $I_n(120,210)$. Then we use the kernel method, obtain generating functions
of each class, and find enumerating formulas. Lin and Yan studied the
classification of the Wilf-equivalences for inversion sequences avoiding pairs
of length-three patterns and showed that there are 48 Wilf classes among 78
pairs. In this paper, we solve six open cases for such pattern classes.
