What is the asymptotical bound of this recurrence relation? I have the recurrence relation, with two initial conditions
$$T(n) = T(n-1) + T(n-2) + O(1)$$
$$T(0) = 1,  \qquad T(1) = 1$$
With the help of Wolfram Alpha, I managed to get the result of $O(\Phi^n)$, where $\Phi = \frac{1+\sqrt 5}{2} \approx 1.618$ is the golden ratio.
Is this correct and how can that be mathematically proven?

You have essentially stated the Fibonacci sequence, or at least asymptotically.  There are numberless references, here for instance.  And your result is not correct; as you will see from the reference, the Fib sequence behaves as $\phi^n/\sqrt{5}$
