The Fibonacci sequence is a famous series of numbers in which each value is created by adding the two preceding numbers. The sequence begins with:

• $F(0) = 0$

• $F(1) = 1$

From that point on, every term is defined as:

• $F(n) = F(n - 1) + F(n - 2)$, for any $n$ greater than $1$

Your task is to compute the value of $F(n)$ for a given integer n.

Constraints:

• $0 \leq$ n $\leq 30$

Before diving into the optimized solution, it’s worth noting that a straightforward recursive solution exists for this problem, but it is extremely slow and can quickly overflow the recursion stack for large inputs. A dynamic programming version improves performance and avoids stack issues, but it still runs in linear time. By using mathematical properties of the Fibonacci sequence, however, we can devise a solution that runs in logarithmic time while using constant space.

The Fibonacci sequence has a surprising connection to a famous mathematical constant called the golden ratio, denoted as $\phi$ (phi). Mathematicians discovered that each Fibonacci number can be approximated using a closed-form expression known as Binet’s formula. This formula shows that instead of building the sequence step by step, you can jump directly to the $n^{th}$ ...