The objective is to find what the next number is.
0, 1, 1, 0, 2, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 0, 0, 1, 1, 2, 2, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 0, 0, 1, 0, 2, 1, 0, 2, 0, 0, 0, 0, 2, 1, 0, 2, 0, 0, 0, 0, 2, 1, 0, 2, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 0, 0, 1, 1, 2, 2, 0, 0, 0, 0, 1, 0, 2, 1, 0, 0, 0, 0, 1, 0, 2, 1, 0, 2, 0, 0, 1, 0, 2, 1, 0, 2, 0, 0, 0, 0, 2, 1, 0, 2, 0, 0, 0, 0, 1, 1, 2, 2, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 0, 0, 1, 0, 2, 1...
After every day, a number gets added onto the end of the list (it's either a 0, 1 or a 2). So the only way to know if we're right is to wait till the next day...
Anybody have any ideas as to how to approach this? It seems there's a pattern but it just hurts me.
Edit: Context is as follows...
In a city there are 2 houses. House 1 is north, and house 2 is south.
The owners of those houses might become forgetful and leave their house for the day with their door unlocked.
0 represents that both doors are locked (they are not accessible).
1 represents the northern house is unlocked. 2 represents the southern house is unlocked.
Only 1 house can be unlocked at a time. Either one house is unlocked, or both are not.
Edit2: 23/02/2018: 0
Edit3: 24/02/2018: 2
Edit4: 25/02/2018: 1
(This is a new tentative answer.)
As explained by the OP, the numbers are generated daily by computer. It seems the first $8\times9 = 72$ numbers, labeled by columns $1,2,3,\dots8$ are "copied" by the next $72$ numbers with "shifted" columns $4,\color{blueviolet}1,6,\color{blueviolet}3,8,\color{blueviolet}5,2,\color{blueviolet}7$.
For example, column $4$ has $0,0,2,2,2,2,2$ :
$$\begin{array}{|c|c|} \hline {\color{red}{1,2,3,4,5,6,7,8,}\\ \color{blue}{0, 1, 1, 0, 2, 0, 0, 0,\\ 0, 1, 1, 0, 2, 0, 0, 0,}\\ 0, 1, 1, 2, 2, 0, 0, 0,\\ 0, 1, 0, 2, 2, 0, 0, 0,\\ 0, 1, 0, 2, 1, 0, 2, 0,\\ \color{blue}{0, 0, 0, 2, 1, 0, 2, 0,\\ 0, 0, 0, 2, 1, 0, 2, 0,}} &{\color{red}{4,1,6,3,8,5,2,7,}\\ \color{blue}{0, 0, 0, 1, 0, 2, 1, 0,\\ 0, 0, 0, 1, 0, 2, 1, 0,}\\ 2, 0, 0, 1, 0, 2, 1, 0,\\ 2, 0, 0, 0, 0, 2, 1, 0,\\ 2, 0, 0, 0, 0, 1, 1, 2,\\ \color{blue}{2, 0, 0, 0, 0, 1, 0, 2,\\ 2, 0, 0, 0, 0, 1,}\color{brown}{0,2,}}\\ \hline 0, 0, 0, 1, 1, 0, 2, 0,\\ 0, 0, 0, 1, 1, 2, 2, 0,\\ \hline \end{array}$$
Thus, my guess is for the next two days, the next two numbers will be $0,2$ (in brown). This is a testable hypothesis, assuming the OP will come back with more data.
(Old answer.)
It's not completely random, but if it does have a pattern, it's tricky. Using the repeating $0,0,0,0$ as a place-holder, we have,
$$?,?,?,?,?,?,?,0,\\ 1, 1, 0, 2, 0, 0, 0, 0,\\ 1, 1, 0, 2, 0, 0, 0, 0,\\ \color{blue}{1, 1, 2, 2, 0, 0, 0, 0,\\ 1, 0, 2, 2, 0, 0, 0, 0,}\\ \color{brown}{1, 0, 2, 1, 0, 2, 0, 0, 0, 0,}\\ 2, 1, 0, 2, 0, 0, 0, 0,\\ 2, 1, 0, 2, 0, 0, 0, 0,\\ 1, 1, 0, 2, 0, 0, 0, 0,\\ \color{blue}{1, 1, 2, 2, 0, 0, 0, 0,\\ 1, 0, 2, 1, 0, 0, 0, 0,}\\ \color{brown}{1, 0, 2, 1, 0, 2, 0, 0, 1, 0,}\\ 2, 1, 0, 2, 0, 0, 0, 0,\\ 2, 1, 0, 2, 0, 0, 0, 0,\\ \color{blue}{1, 1, 2, 2, 0, 0, 0, 0,\\ 1, 0, 2, 2, 0, 0, 0, 0,}\\ 1,?,?,?,?,?,?,?,$$