1989 Putnam A6

Let $\alpha=1+a_1x+a_2x^2+\cdots$ be a formal power series with coefficients

in the field of two elements. Let

$$

a_n =

\begin{cases}

1 & \parbox{2in}{if every block of zeros in the binary expansion of $n$

has an even number of zeros in the block} \\[.3in]

0 & \text{otherwise.}

\end{cases}

$$

(For example, $a_{36}=1$ because $36=100100_2$ and $a_{20}=0$ because

$20=10100_2.$)

Prove that $\alpha^3+x\alpha+1=0.$

令 $\alpha=1+a_1x+a_2x^2+\cdots$ 为带有系数的形式幂级数

在二元领域。让

$$

a_n =

\开始{案例}

1 & \parbox{2in}{if $n$ 的二进制展开式中的每个零块

块中有偶数个零} \\[.3in]

0 & \text{否则。}

\结束{案例}

$$

(例如，$a_{36}=1$ 因为 $36=100100_2$ 且 $a_{20}=0$ 因为

$20=10100_2.$)

证明 $\alpha^3+x\alpha+1=0.$