Narayana sequence and the Brocard-Ramanujan equation

Let {an}n & GE;0 be the Narayana sequence defined by the recurrence an = an-1 + an-3 for all n & GE; 3 with intital values a0 = 0 and a1 = a2 = 1. In this paper, we fully characterize the 3-adic valuation of an + 1 and an - 1 and then we find all positive integer solutions (u, m) to the Brocard-Ramanujan equation m! + 1 = u2 where u is a Narayana number.