1999 Putnam 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

$$a_n^2 + a_{n+1}^2 = a_m .$$

考虑幂级数展开

$$\frac{1}{1-2x-x^2} = \sum_{n=0}^\infty a_n x^n。$$

证明，对于每个整数 $n\geq 0$，都有一个整数 $m$ 使得

$$a_n^2 + a_{n+1}^2 = a_m .$$