ILL 1977 Problem 42

The sequence $a_{n,k} \ , k = 1, 2, 3,\ldots, 2^n \ , n = 0, 1, 2,\ldots,$ is defined by the following recurrence formula: $$a_1 = 2, a_{n,k} = 2a_{n-1,k}^3, , a_{n,k+2^{n-1}} =\frac 12 a_{n-1,k}^3$$$$\text{for} k = 1, 2, 3,\ldots, 2^{n-1} \ , n = 0, 1, 2,\ldots$$ Prove that the numbers $a_{n,k}$ are all different.