  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's
friezes. Their number and their properties are closely linked to the structure
and the cardinality of the chosen set. The main objective of this text is to
obtain an explicit formula giving the number of $\lambda$-quiddities of odd
size, and a lower and upper bound for the number of $\lambda$-quiddities of
even size, over the rings $\mathbb{Z}/2^{m}\mathbb{Z}$ ($m \geq 2$). We also
give explicit formulas concerning the number of $\lambda$-quiddities of size
$n$ over $\mathbb{Z}/8\mathbb{Z}$.
