I have a reccurence relation that relates to a keystream in a Linear-Feedback Shift Register, working in Mod 3 so I'm using digits ${0,1,2}$
$Z_{i+2}=C_0Z_i+C_1Z_{i+1}$
I have apart of the keystream $S=...11022...$
I am looking to find $C_0$ & $C_1$ along with
the three keystream numbers that follow and precede $S$
Heres what I have so far, I set up a system of equations
$1*C_0 + 1*C_1=0$ which implies $C_0=-C_1$
$1*C_0 + 0*C_1=2$
I solve the system of equations to get $C_0 = 1$ and $c_1=2$
But If I plug those back into my second equation, it still doesn't satisfy my system. I feel like I'm missing something extremely simple or making a very simple algebra mistake.
When you solved for $C_0,C_1$ you made an error. $1*C_0+0*C_1=2$ is not satisfied by $C_0=1, C_1=2$. The correct solution is $C_0=2,C_1=1$, which then produces $2*0+1*2=2$ for the last position of your string.