On compositions whose parts are polygonal numbers

Abstract. For a given strictly increasing sequence \(\{a_n\}\) of natural numbers, let \(g(n)\) be the number of compositions of \(n\) all of whose parts belong to \(\{a_n\}\). We derive a recurrence that enables the computation of \(g(n)\), as well as an estimate for \(g(n)\) for large \(n\). We use these results to obtain useful data concerning compositions whose parts belong to the following 3 sequences: triangular, square, and pentagonal numbers.