  Zeckendorf's theorem states every positive integer has a unique decomposition
as a sum of non-adjacent Fibonacci numbers. This result has been generalized to
many sequences $\{a_n\}$ arising from an integer positive linear recurrence,
each of which has a corresponding notion of a legal decomposition. Previous
work proved the number of summands in decompositions of $m \in [a_n, a_{n+1})$
becomes normally distributed as $n\to\infty$, and the individual gap measures
associated to each $m$ converge to geometric random variables, when the leading
coefficient in the recurrence is positive. We explore what happens when this
assumption is removed in two special sequences. In one we regain all previous
results, including unique decomposition; in the other the number of legal
decompositions exponentially grows and the natural choice for the legal
decomposition (the greedy algorithm) only works approximately 92.6\% of the
time (though a slight modification always works). We find a connection between
the two sequences, which explains why the distribution of the number of
summands and gaps between summands behave the same in the two examples. In the
course of our investigations we found a new perspective on dealing with roots
of polynomials associated to the characteristic polynomials. This allows us to
remove the need for the detailed technical analysis of their properties which
greatly complicated the proofs of many earlier results in the subject, as well
as handle new cases beyond the reach of existing techniques.
