Mathematical explorations on the sequence of factoriangular numbers: Extending the results on generalizations

A factoriangular number is formed by adding a factorial and a triangular number. If corresponding factorials and triangular numbers are added, the results are n-factoriangular numbers. Other factoriangular numbers are called (n,k)-factoriangular numbers, n^(m)-factoriangular numbers, (n^(m),k^(m))-factoriangular numbers, and (n^(a),k^(b))-factoriangular numbers. The main objective of this study is to explore the sequence of (n^(m),k^(m))-factoriangular numbers and the sequence of (n^(a),k^(b))-factoriangular numbers as generalizations of the sequence of n-factoriangular numbers. This research is a discipline-based scholarship of discovery that employs an exploratory method involving the scientific approach of experimental mathematics. The mathematical method was used in doing the explorations, focusing on the formulations and proofs of theorems and giving some examples. For the main results, ten theorems were proven and several examples of sequences were provided. The theorems include several formulas for (n^(m),k^(m))-factoriangular numbers, and (n^(a),k^(b))-factoriangular numbers. The proofs for theorems in (n^(m),k^(m))-factoriangular numbers are applicable for similar theorems in (n^(a),k^(b))-factoriangular numbers. Specific sequences of some generalized factoriangular numbers were presented in tables. Entries of numbers in the tables may lead to the formation of triangular arrays of factoriangular numbers that may be further explored by other researchers, especially those mostly interested in recreational mathematics.