And now your memory usage will grow eye-wateringly large. Instead, convert the algorithm to be iterative or at least tail-recursive and it will be faster than both the naive and memoized versions!
Or use the closed-form solution.
The "closed-form solution" is slower than standard method. It just uses arbitrary-precision fractional arithmetic (square root, exponentiation, division) instead of arbitrary-precision integer arithmetic (exponentiation of a 2x2 integer matrix).
The best solution is to use the O(log n) time exponentiation of a matrix, which is fast enough to be constant.