I've encountered this question, asking to solve the recurrence $$F(n,m) = F(n-1,m) + F(n,m-1)$$ for some initial conditions. I wonder how the solution would change if we add a constant, say
$ G(n,m) = G(n-1,m) + G(n,m-1)+2$.
One answer for the mentioned post suggested generating functions. What would change here? For simplicity, let's assume $G(1,m) = m − 1$ and $G(n, 1) = 0$.