Decomposition by subtraction In how many ways one can decompose an integer $n$ to smaller integers at least 3? for example 13 has the following decompositions:
\begin{gather*}
13\\
3,10\\
4,9\\
5,8\\
6,7\\
3,3,7\\
3,4,6\\
3,5,5\\
3,3,3,4\\
4,4,5\\
\end{gather*} 
Points and hints are welcome.

Let $p(k,n)$ be the number of partitions of $n$ into parts at least as large as $k$ then
$$p(k,n) = p(k,n-k) + p(k+1,n)$$
as per Partition (Number Theory).
And so
$$ p(k+1,n) = p(k,n) - p(k,n-k).$$
Therefore
$$ \begin{align}
p(3,n) &= p(2,n) - p(2,n-2) \\
&= \{ p(1,n) - p(1,n-1) \} - \{ p(1,n-2) - p(1,n-3) \} \\
&= p(n) - p(n-1) - p(n-2) + p(n-3),
\end{align} $$
where $p(n)$ is the usual partition function, as per quantumelixir's formula.
