Putnam 1999 A3

Consider the power series expansion $$\frac{1}{1-2x-x^2} = \sum_{n=0}^\infty a_n x^n.$$ Prove that, for each integer $n\geq 0$, there is an integer $m$ such that $$a_n^2 + a_{n+1}^2 = a_m .$$