Linear combinations of factorials in binary recurrence sequences

Abstract

Let $\{u_{n}\}_{n \geq 0}$ be a non-degenerate binary recurrence sequence with positive discriminant. In this paper, we consider the Diophantine equation $u_m + u_n = a_1 n_1! + \cdots + a_k n_k!$ and prove that there are only finitely many effectively computable terms which can be expressed as a sum of factorials. Furthermore, we find the terms of the balancing sequences that can be written as a sum of two factorials.