Find two infinite subsets A and B of $N$ and $0$ such that every natural number can be written uniquely as the sum of a number in A and a number in B. Recently encountered the following question:

Find two infinite subsets $A$ and $B$ of $N$ (including $0$) such that every natural number can be written uniquely as the sum of a number in $A$ and a number in $B$.

I have tried finding some sets and so far have only found the following:
$A = [1,3,5,7,9,11...]$
$B = [0,1,2,3,4,5...]$
Is there a way to find a function possibly for all the values of the subset? How would you attack this question, would you use brute force or some other tool? If so, what? And if possible can you please post your solution to the problem.
Thanks heaps.

You can choose sums of distinct powers of 4 for the set A ($\{0, 1, 4, 5, 16, 17, 20, ...\}$), and twice the sums of distinct powers of 4 for the set B ($\{0, 2, 8, 10, 32, 34, 40, ...\}$).
The sets A and B correspond to the OEIS sequences A000695 and A062880, respectively.
The underlying idea is to separate even and odd indexed bits in binary expansions of a number $n$ to get the unique element of A and of B that sum to that number $n$.
