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) with 1D dp : https://cses.fi/paste/d2e2d0644e0a8603822f62/

tabulation solution (accepted) with 2D dp : https://cses.fi/paste/d40aa6fee8c71199825f90/

memoization solution (TLE) with 2D dp : https://cses.fi/paste/972af1e5f43728b8825fa2/

memoization solution for this problem with 1D dp seems like does no exist.