  Consider the weight $\lambda$ which is the sum of all simple roots of a
simple Lie algebra. Using Kostant's weight multiplicity formula we describe and
enumerate the contributing terms to the multiplicity of the zero weight in the
representation with highest weight $\lambda$. We prove that in Lie algebras of
type $A$ and $B$, the number of contributing terms to the multiplicity of the
zero-weight space in the representation with highest weight $\lambda$ is given
by a Fibonacci number, and that in Lie algebras of type $C$ and $D$, the
analogous result is given by a multiple of a Lucas number.
