  It is well known that every positive integer can be expressed as a sum of
nonconsecutive Fibonacci numbers provided the Fibonacci numbers satisfy $F_n
=F_{n-1}+F_{n-2}$ for $n\geq 3$, $F_1 =1$ and $F_2 =2$. In this paper, for any
$n,m\in\mathbb{N}$ we create a sequence called the $(n,m)$-bin sequence with
which we can define a notion of a legal decomposition for every positive
integer. These sequences are not always positive linear recurrences, which have
been studied in the literature, yet we prove, that like positive linear
recurrences, these decompositions exist and are unique. Moreover, our main
result proves that the distribution of the number of summands used in the
$(n,m)$-bin legal decompositions displays Gaussian behavior.
