  We study the problem of generating interesting integer sequences with a
combinatorial interpretation. For this we introduce a two-step approach. In the
first step, we generate first-order logic sentences which define some
combinatorial objects, e.g., undirected graphs, permutations, matchings etc. In
the second step, we use algorithms for lifted first-order model counting to
generate integer sequences that count the objects encoded by the first-order
logic formulas generated in the first step. For instance, if the first-order
sentence defines permutations then the generated integer sequence is the
sequence of factorial numbers $n!$. We demonstrate that our approach is able to
generate interesting new sequences by showing that a non-negligible fraction of
the automatically generated sequences can actually be found in the Online
Encyclopaedia of Integer Sequences (OEIS) while generating many other similar
sequences which are not present in OEIS and which are potentially interesting.
A key technical contribution of our work is the method for generation of
first-order logic sentences which is able to drastically prune the space of
sentences by discarding large fraction of sentences which would lead to
redundant integer sequences.
