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?
 A: Occasionally (as in the link posted by vadim123) you see "Fibonacci integer sequence".
Lucas sequences (of which the Lucas sequence is but one example) are a slight generalization. Sometimes the term Horadam sequence is used instead.
The general classification under which all of these fall is the linear recurrence relation. Most of the special properties of the Fibonacci sequence are inherited from either linear recurrence relations or divisibility sequences.
A: 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. 
