In a paper written in 2001, Knopfmacher and Mays introduced the concept of graph compositions. In another paper written in 2004, Ridley and Mays proved a theorem on the number of compositions of the Cartesian product of the path graph with another graph. We build on this theorem to create an algorithm that calculates the closed-form solutions for the sequences given by the numbers of compositions of the Cartesian product of the path graph with complete graphs, and we apply this algorithm to produce two new sequences in the On-Line Encyclopedia of Integer Sequences. We also demonstrate some unexpected results that arise from these sequences, and try to explain why they might occur.