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How to use $|n|$ to generate a sequence

Can someone explain the basics of the sequence $a_n$ where $a_n = -1 + \left|\frac{n}{2}\right|$?
I know that it’s some form of the Fibonacci sequence, but is there a way to derive its formula, and a way to stop the expansion without taking many terms?
For example, how does one derive $a_{n+2} – 2a_{n+1} + a_n = 0$ from $a_n = -1 + |\frac{n}{2}|$?

A:

By definition
$$a_n=a_{n-1}-\left|\frac{n-1}{2}\right|$$
which simplifies to
$$a_n=a_{n-2}+\left|\frac{n-1}{2}\right|$$
Where $a_0=0$. Now the general term is
$$a_n=a_{n-r}+\left|\frac{n-1}{2}\right|$$
Now set $r=2$ to get your $a_{n+2}-2a_{n+1}+a_n=0$

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