How to find the union of any progression?

Question

Here is an answer to find the union of two arithmetic progression. How to find a general formula for union of two arithmetic progressions

But, is there a formula to find the union of the sets of two non-arithmetic, non-geometric progressions?

For example,the union of $ {2^n}$ and $ \frac{2^n - 1}{3}$ ?

By union, I mean the union of two sets like if the progressions are 2n(even numbers) and 2n+1 (odd numbers) then the union would be n (whole numbers)

If progressions are

P1 = {0,2,4,6,...} and

P2 = {1,3,5,7,...}

Then the union

$ P_1 \space U \space P_2 = {0,1,2,3,4,5,6,7,..} $