Putnam 1989 A6

Let $\alpha=1+a_1x+a_2x^2+\cdots$ be a formal power series with coefficients in the field of two elements. Let $$a_n = \begin{cases} 1 & \text{if every block of zeros in the binary expansion of $n$ has an even number of zeros in the block} \\[.3in] 0 & \text{otherwise.} \end{cases}$$ (For example, $a_{36}=1$ because $36=100100_2$ and $a_{20}=0$ because $20=10100_2.$) Prove that $\alpha^3+x\alpha+1=0.$