Let $b(n)$ be the binary partition function of $n$, i.e., the number of partition of $n$ for which all parts are the powers of two, and have properties $b(2n+1)=b(2n)$ and $b(2n)=b(n)+b(2n-2)$, for all positive integer $n$, where $b(0)=1$. Let $a(n)$ be $1$ if the number of zeros in the binary expression of $n$ is even, and $-1$ if the number of zeros in the binary expression of $n$ is odd.
Prove that $b(2n) \equiv 1+a(n) \pmod 4$ for all $n \ge 1$.
I tried for small values of $n$, but still didn't get any idea, even if a pattern. How to show it correctly? My intuition is said that it can be proved by using the generating function. But, I don't know yet the generating function of $a(n)$.