Computing Fibonacci Numbers

Let's call $T(n)$ the number of arithmetic operations we need to compute $F(n)$. $T(0) = T(1) = 0$, because those values are given to us. $T(2) = 1$, because we need to add $F(0)$ and $F(1)$. $T(3) = 2$, because we need to compute $F(2)$ and then add it to $F(1)$. In general, we have that $T(n) = T(n-1) + T(n-2) + 1$: we need to compute the two numbers below the one we want and then add them together. The $T(n)$ sequence bears a striking resemblance to the Fibonacci sequence if you compare the definitions—this is a good example of how the Fibonacci sequence or similar sequences pop up in analyzing recursive algorithms.¹

I've graphed $T(n)$ for $n$ from 0 to 50. Note the scale on the y-axis, from 0 to 20 billion! $T_n$ grows exponentially, and gets really big really fast. Clearly this won't be a viable way of computing large Fibonacci numbers.