What is the Broader Name for the Fibonacci Sequence and the Sequence of Lucas Numbers? Fibonacci and Lucas sequences are very similar in their definition. However, I could just as easily make another series with a similar definition; an example would be:
$$x_0 = 53$$
$$x_1 = 62$$
$$x_n = x_{n - 1} + x_{n - 2}$$
What I want to ask is, what is the general name for these types of sequences, where one term is the sum of the previous two terms?

The Fibonacci sequence is a member of the "broader'' family of sequences known as the Lucas sequences, $U_{n}(P, Q)$, recursively defined by
$$ x_{n} = Px_{n-1} - Qx_{n-2}$$
with $P = 1$ and $Q = -1$.
Similarly, the sequence of Lucas numbers comes from the associated family of companion Lucas sequences, $V_{n}(P, Q)$, which are recursively defined the same as the Lucas sequences.  
Explicitly, it follows that
$$ U_{n} = \frac{{\phi}^{n} - {\theta}^{n}}{\phi - \theta}$$
and
$$ V_{n} = {\phi}^{n} + {\theta}^{n},$$
where $\phi$ and $\theta$ are the roots of $x^{2} - Px + Q = 0$, the characteristic equation associated with the aforementioned difference equation. 
