lets split that set in two series: $ a_i = 2^{2i}3 = 4^i 3$ for $ i \ge 0$ and $ b_i = - 2^{2i + 1} = - 4^i 2$ for $ i \ge 0$ Defining: $ c_i = a_i + b_i = 4^i$ for $ i \ge 0$ $ d_i = b_i = - 2 \times 4^i$ for $ i \ge 0$ $ e_i = a_i = 3 \times 4^i$ for $ i \ge 0$ Every positive integer can be written as a sum of powers of 4 using the digits $ \{0,1, - 2,3\}$ by replacing in finite steps the pair of digits $ \cdots (x)(2) \cdots$ with $ \cdots (x + 1)( - 2) \cdots$ then every positive integer is equal to the sum of distinct numbers of the original set.