On a variant of the Brocard—Ramanujan equation and an application

In this paper, we study the variant of the Brocard—Ramanujan diophantine equation $m!+1=u^2$, where $u$ is a member of a sequence of positive integers. Under some technical conditions on the sequence, we prove that this equation has at most finitely many solutions in positive integers $m$ and $u$. As an application, we completely solve this equation when $u$ is a Tripell number. The Tripell numbers are defined by the recurrence relation $T_n=2T_{n-1}+T_{n-2}+T_{n-3}$ for $n\geq 3$, with $T_0=0$, $T_{1}=1$ and $T_2=2$ as initial conditions.