This is a bit of a soft question, but I hope it's still precise enough. I'm interested in finding cases where a complicated expression has a simple approximation which becomes exact again upon rounding / flooring / ceiling. Examples follow.
An exact expression for the number of derangements is $$ D_n = n!\sum_{k=0}^n\frac{(-1)^k}{k!}. $$ The sum here is very close to $e^{-1}$, and in fact the following formula also holds: $$ D_n=\left[\frac{n!}{e}\right] $$ where the brackets denote rounding to the nearest integer.
There are even nicer examples which involve only rational numbers and unify different cases of a problem. For example, the number of triangles in a triangular matchstick arrangement can be expressed either by the case-by-case formula $$ \binom{n+2}3+\begin{cases} \frac1{24}n(n+2)(2n-1),&\text{if }n\text{ is even}\\\\ \frac1{24}\left(n^2-1\right)(2n+3),&\text{if }n\text{ is odd} \end{cases} $$ or by the floored expression $$ \left\lfloor\frac{n(n+2)(2n+1)}8\right\rfloor. $$ A second example of this kind would be counting the number of triangles inscribed in an $n$-gon up to rotations and reflections which (if I'm not mistaken) comes out to $$ \frac{(n-1)(n-2)}{12} + \begin{cases} \frac{n-2}{4},&\text{if }n\text{ is even}\\\\ \frac{n-1}{4},&\text{if }n\text{ is odd} \end{cases}+\begin{cases} \frac{1}{3},&\text{if }n\text{ is divisible by }3\\\\ 0,&\text{otherwise} \end{cases} $$ and which is also equal to $$ \left[\frac{n^2}{12}\right]. $$ What are some other instances of this phenomenon?
