Distribution between $0$ and $n$ with closed for for expected value conditional on being greater than x.

Question

I'm in the market for a distribution. Should be between $0$ and $n$. Preferably, but not necessarily over the integers. And most importantly, the expected value conditional on it being greater than $x$ should have a nice closed form. Does such a distribution exist? If we didn't need it to have an upper bound, we could have used the geometric distribution with its memory-less property.