Is there any case where memoization approach requires more states than tabulation?

I have heard that tabulation and memoization only differs by some memory efficiency and time efficiency.

But now I am facing a problem where the tabulation solution only needs a state, "amount". I tried to implement it using memoization but there I must keep track of the "index", otherwise it gives wrong answer.

problem link: https://cses.fi/problemset/task/1636/

tabulation solution (accepted) : https://cses.fi/paste/d2e2d0644e0a8603822f62/

And it is clear that memoization not only needs extra state here but also is impossible with the given constraints. Because it needs a 2D dp array with n * x size, so I do not have any submission link for memoization.