Combinations with a condition of limited length If there was a mold of size $N$ and $N$ uniquely sized bricks $(N, N-1,\cdots, 1)$, in how many ways the bricks could be places into the mold? The order doesn't matter.
For examples, for $N = 5$ there are $9$ possible combinations:
$[5], [4+1], [4], [3+2], [3+1], [3], [2+1], [2], [1].$
Note: duplicates (e.g. [2+2], [1+1]) are not allowed because all the bricks are unique. Here are some more combinations with small N: $1 \to 1, 2 \to 2, 3 \to 4, 4 \to 6, 5 \to 9, 6 \to 13, 7 \to 18, 8 \to 24, 9 \to 32$...

To answer this question, start by considering how many combinations there are of non-negative integers $n$ and $m$ such that $n+m=N$:
If $N$ is even we have $\frac{N+2}{2}$ combinations. If $N$ is odd we have $\frac{N+1}{2}$ combinations. 
To get all possible combinations sum together the result above for all numbers going from $N$ to $1$: 
If $N$ is even we have 
\begin{align}C_{even}&=\frac{N+2}{2}+\frac{N-1+1}{2}+\frac{N-2+2}{2}+\frac{N-3+1}{2}+\cdots+\frac{2+2}{2}+\frac{1+1}{2}\\
&= \frac{1}{2}\left(\frac{(N+1)}{2}\times N+\frac{N}{2}\times 2+\frac{N}{2}\times 1\right)\\
&= \frac{N(N+1)+3N}{4}.
\end{align}
If $N$ is odd we find 
$$C_{odd}= \frac{N(N+1)+3N-1}{4}.$$
