  A beautiful theorem of Zeckendorf states that every positive integer can be
uniquely decomposed as a sum of non-consecutive Fibonacci numbers $\{F_n\}$,
where $F_1 = 1$, $F_2 = 2$ and $F_{n+1} = F_n + F_{n-1}$. For general
recurrences $\{G_n\}$ with non-negative coefficients, there is a notion of a
legal decomposition which again leads to a unique representation, and the
number of summands in the representations of uniformly randomly chosen $m \in
[G_n, G_{n+1})$ converges to a normal distribution as $n \to \infty$.
  We consider the converse question: given a notion of legal decomposition, is
it possible to construct a sequence $\{a_n\}$ such that every positive integer
can be decomposed as a sum of terms from the sequence? We encode a notion of
legal decomposition as a function $f:\N_0\to\N_0$ and say that if $a_n$ is in
an "$f$-decomposition", then the decomposition cannot contain the $f(n)$ terms
immediately before $a_n$ in the sequence; special choices of $f$ yield many
well known decompositions (including base-$b$, Zeckendorf and factorial). We
prove that for any $f:\N_0\to\N_0$, there exists a sequence
$\{a_n\}_{n=0}^\infty$ such that every positive integer has a unique
$f$-decomposition using $\{a_n\}$. Further, if $f$ is periodic, then the unique
increasing sequence $\{a_n\}$ that corresponds to $f$ satisfies a linear
recurrence relation. Previous research only handled recurrence relations with
no negative coefficients. We find a function $f$ that yields a sequence that
cannot be described by such a recurrence relation. Finally, for a class of
functions $f$, we prove that the number of summands in the $f$-decomposition of
integers between two consecutive terms of the sequence converges to a normal
distribution.
