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For enumerative problems, i.e. computable functions f from N to Z, we definethe notion of an effective (or closed) formula. It is an algorithm computingf(n) in the number of steps that is polynomial in the combined size of theinput n and the output f(n), both written in binary notation. We discuss manyexamples of enumerative problems for which such closed formulas are, or arenot, known. These problems include (i) linear recurrence sequences andholonomic sequences, (ii) integer partitions, (iii) pattern-avoidingpermutations, (iv) triangle-free graphs and (v) regular graphs. In part I wediscuss problems (i) and (ii) and defer (iii)--(v) to part II. Besides otherresults, we prove here that every linear recurrence sequence of integers has aneffective formula in our sense.

What is an answer? - remarks, results and problems on PIO formulas in combinatorial enumeration, part I