RMM 2017 Problem 1

(a) Prove that every positive integer $n$ can be written uniquely in the form

$$n=\sum_{j=1}^{2 k+1}(-1)^{j-1} 2^{m_{j}}$$

where $k \geq 0$ and $0 \leq m_{1}<m_{2}<\cdots<m_{2 k+1}$ are integers. This number $k$ is called the weight of $n$. (b) Find (in closed form) the difference between the number of positive integers at most $2^{2017}$ with even weight and the number of positive integers at most $2^{2017}$ with odd weight.