  Berenshtein and Zelevinskii provided an exhaustive list of pairs of weights
$(\lambda,\mu)$ of simple Lie algebras $\mathfrak{g}$ (up to Dynkin diagram
isomorphism) for which the multiplicity of the weight $\mu$ in the
representation of $\mathfrak{g}$ with highest weight $\lambda$ is equal to one.
Using Kostant's weight multiplicity formula we describe and enumerate the
contributing terms to the multiplicity for subsets of these pairs of weights
and show that, in these cases, the cardinality of these contributing sets is
enumerated by (multiples of) Fibonacci numbers. We conclude by using these
results to compute the associated $q$-multiplicity for the pairs of weights
considered, and conjecture that in all cases the $q$-multiplicity of such pairs
of weights is given by a power of $q$.
