The unexpected, $\ 2^n$ is not of the form $2xy + x$?

Question

When testing polynomials for items of an integer sequence, I find that $2^n$ is not of the form $2xy + x,\ n,\ x,\ y \in \mathbb N.\ $How to prove this?

Answer 1

We have $2xy+x=(2y+1)x$, and $2y+1$ is odd. The only positive odd divisor of $2^n$ is $1$. It follows that we must have $y=0$, which presumably you don't allow.