number theory - Prove that $sum_{i=0}^{63}f_{i}cdotleft(n+iright)^{5}=0$

For $n \geq 1$, let $f_m = (-1)^s$ where $s$ is the digital sum modulo $2$ of the binary representation of $m$. Prove that

$$\sum_{i=0}^{63}f_{i}\cdot\left(n+i\right)^{5}=0.$$

Since $s$ is the digital sum taken modulo $2$ we know that $s \in \{0,1\}$. I don't see a pattern in digital sum of the binary representation modulo $2$: $0,1,1,0,1,0,0,1,1,0,0,1,0,1,\ldots$. How do we prove the sum is equal to zero?