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.

https://cses.fi/problemset/result/8535002/

And it is clear that memoization not only needs extra state here but also is impossible with the given constraints.