It is well known that every positive integer can be expressed as a sum of
nonconsecutive Fibonacci numbers provided the Fibonacci numbers satisfy
for
,
and
. For any
we create a sequence
called the
-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
-bin
legal decompositions displays Gaussian behavior.
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