  The $\lambda$-quiddities of size $n$ are $n$-tuples of elements of a fixed
set, solutions of a matrix equation appearing in the study of Coxeter friezes.
These can be considered on various sets with very different structures from one
set to another. The main objective of this text is to obtain explicit formulas
giving the number of $\lambda$-quiddities of size $n$ over finite fields and
over the rings $\mathbb{Z}/N\mathbb{Z}$ with $N=4m$ and $m$ square free. We
will also give some elements about the asymptotic behavior of the number of
$\lambda$-quiddities verifying an irreducibility condition over
$\mathbb{Z}/N\mathbb{Z}$ when $N$ goes to the infinity.
