IT-CESENATICO 1987 Problem 4

Given $I_0 = \{-1,1\}$ , define $I_n$ recurrently as the set of solutions $x$ of the equations $x^2 -2xy+y^2- 4^n = 0$ , where $y$ ranges over all elements of $I_{n-1}$ . Determine the union of the sets $I_n$ over all nonnegative integers $n$ .