When is a number not of a form below a certain input? x*y + y + x +1 with x < c

Question

Numbers have certain forms. Like $3k + 1$ applies to 1,4,7 so on. I'm interested in how to characterize what numbers are not of a particular form. For example, $3k$ and $3k + 2$ numbers are not $3k + 1$. In addition, it would be helpful if I could set a limit on a variable's value. As an arbitrary expression (I don't care about solutions to this one in particular), say we have $xy + y + x +1$ with $x < c$. How can I easily generate all the numbers that are not of that form? If $c = 4$ then that would be anything that is not $2y + 2$, $3y + 3$ or $4y + 4$ or in other words anything that is not $(x+1)y + x + 1$ where $x < c$, but that does not particular help me as I can't plug in an arbitrary value without checking the other forms (say I plug in $0$ for $y$ in the $2y + 2$ equation and then add $1$ to make it not of that form. Then $3y + 3$ with $y = 0$ covers that number anyway). I could code this for particular examples, but I'm looking to understand how to do this generally. Thank you for any help!