Putnam 1976 A2

A-2. Let $P(x, y)=x^{2} y+x y^{2}$ and $Q(x, y)=x^{2}+x y+y^{2}$. For $n=1,2,3, \ldots$, let $F_{n}(x, y)=(x+y)^{n}-x^{n}-y^{n}$ and $G_{n}(x, y)=(x+y)^{n}+x^{n}+y^{n}$. One observes that $G_{2}=2 Q, F_{3}=3 P, G_{4}=2 Q^{2}, F_{5}=5 P Q, G_{6}=2 Q^{3}+3 P^{2}$. Prove that, in fact, for each $n$ either $F_{n}$ or $G_{n}$ is expressible as a polynomial in $P$ and $Q$ with integer coefficients.