  Let $\Lambda^r_n$ be the path algebra of the linearly oriented quiver of type
$\mathbb{A}$ with $n$ vertices modulo the $r$-th power of the radical, and let
$\widetilde{\Lambda}^r_n$ be the path algebra of the cyclically oriented quiver
of type $\widetilde{\mathbb{A}}$ with $n$ vertices modulo the $r$-th power of
the radical. Adachi gave a recurrence relation for the number of $\tau$-tilting
modules over $\Lambda^r_n$. In this paper, we show that the same recurrence
relation also holds for the number of $\tau$-tilting modules over
$\widetilde{\Lambda}^r_n$. As an application, we give a new proof for a result
by Asai on recurrence formulae for the number of support $\tau$-tilting modules
over $\Lambda^r_n$ and $\widetilde{\Lambda}^r_n$.
