  The aim of this article is to obtain a formula giving, for a positive integer
$n$, the number of roots of the $n^{th}$ continuant polynomial over a finite
local ring. In particular, we will give counting formulae for the roots of the
continuant over the local rings $\mathbb{F}_{q}$, $\mathbb{Z}/p^{m}\mathbb{Z}$
and $\frac{\mathbb{F}_{q}[X]}{<X^{m}>}$. To conclude, the methods used for the
continuant will allow us to give a new and short proof of the counting formulae
for $\lambda$-quiddities (which are the solutions of a matrix equation
appearing in the study of Coxeter's friezes) over the rings
$\mathbb{Z}/p^{m}\mathbb{Z}$.
