A note on explicit formulas for Bernoulli polynomials

For $r\in\left \{1,-1,\frac{1}{2}\right\}$, we prove several explicit formulas for the $n$-th Bernoulli polynomial $B_{n}\left(x \right)$, in which $B_{n}\left(x\right)$ is equal to a linear combination of the polynomials $x^{n}$, $\left(x+r\right)^{n},\ldots,$ $\left(x+rm\right)^{n}$, where $m$ is any fixed positive integer greater than or equal to $n$.