Asking about series where the next term is defined by the last one, I don't know what terms to google nor a book to read about them. I read this problem and I don't know how to solve it, nor do I know what to google to learn more about it. 

If $a_0 =0, a_1 = 1$ and $a_{n+1} = 1 + \frac{a_n^2 - a_n a_{n-1} + 1}{a_n - a_{n-1}}$ for $n \geq 1$, what integer is closes to $a_{12}$ 

Could someone tell me how to classify this type of problem and explain the methods of attacking these types of problems. 
 A: These are called Recurrence Relations. You can read more about them here
There are many methods for tackling these, from generating functions to eigenvalue decomposition. You can also try substituting something that appears to be the correct form with variable coefficients. There are many methods.
Sometimes you will see these referred to as Difference Equations (they can be thought of as discrete analogues to differential equations).
A: This one benefits from some arranging. we have
$$ a_{n+1} - a_n = 1 + \frac{1}{a_n - a_{n-1} }. $$
This suggests simply defining $$b_n = a_n - a_{n-1},$$ so that
$$ b_{n+1} = 1 + \frac{1}{b_n} $$
beginning with $b_1 = 1.$ The values of $b_n$ are
$$ 1, 2, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21, $$
and so on. Denominator and numerator are consecutive Fibonacci numbers,
$$ b_n = \frac{F_{n+1}}{F_n}, $$ where we number the Fibonacci numbers so that $F_5 = 5,$ that is how i remember it.
Anyway, $b_n$ is pretty close to $$\phi = \frac{1 + \sqrt 5}{2}$$ and
$a_n - n \phi$ approaches a limit. $a_{12}$ is about $12 \phi \approx 19.416,$ and the closest integer to $a_{12} \approx 19.09796$ is indeed $19.$
There could be an exact formula for $a_n$ based on the exact formula for the Fibonacci numbers. 
