  For enumerative problems, i.e. computable functions f from N to Z, we define
the notion of an effective (or closed) formula. It is an algorithm computing
f(n) in the number of steps that is polynomial in the combined size of the
input n and the output f(n), both written in binary notation. We discuss many
examples of enumerative problems for which such closed formulas are, or are
not, known. These problems include (i) linear recurrence sequences and
holonomic sequences, (ii) integer partitions, (iii) pattern-avoiding
permutations, (iv) triangle-free graphs and (v) regular graphs. In part I we
discuss problems (i) and (ii) and defer (iii)--(v) to part II. Besides other
results, we prove here that every linear recurrence sequence of integers has an
effective formula in our sense.
