The function $f(n)$ is defined for all integers $n$, such that $f(x) + f(y) = f(x + y) - 2xy - 1$ for all integers $x$ and $y$ and $f(1) = 1$ The function $f(n)$ is defined for all integers $n$, such that $f(x) + f(y) = f(x + y) - 2xy - 1$ for all integers $x$ and $y$ and $f(1) = 1$. Find $f(n)$.

I started plugging small values in and I got:
$$f(1)=1$$
$$f(2)=5$$
$$f(3)=11$$
I don't see any pattern so far, and I don't know another way to solve this question.  Solutions are greatly appreciated!

Hint: Let $g(x)=f(x)-x^2$. Then $g(x)+g(y) = g(x+y)-1$ for all $x,y$, so in particular $g(x+1) = g(x)+1$.
