  A beautiful theorem of Zeckendorf states that every integer can be written
uniquely as a sum of non-consecutive Fibonacci numbers
$\{F_n\}_{n=1}^{\infty}$; Lekkerkerker proved that the average number of
summands for integers in $[F_n, F_{n+1})$ is $n/(\phi^2 + 1)$, with $\phi$ the
golden mean. Interestingly, the higher moments seem to have been ignored. We
discuss the proof that the distribution of the number of summands converges to
a Gaussian as $n \to \infty$, and comment on generalizations to related
decompositions. For example, every integer can be written uniquely as a sum of
the $\pm F_n$'s, such that every two terms of the same (opposite) sign differ
in index by at least 4 (3). The distribution of the numbers of positive and
negative summands converges to a bivariate normal with computable, negative
correlation, namely $-(21-2\phi)/(29+2\phi) \approx -0.551058$.
