I have to solve two different homogeneous difference equations.
1) $z_0 = 2 , z_1=1 , z_{j+2} - z_{j+1} -z_j = 0 $
and
2) $z_0 =2 , z_1=0 , z_2 =16 , z_3=-14 ,$
$z_{j+4} - 6z_{j+2} +8z_{j+1} - 3z_j = 0 $
So I solved the first one. I set $z_j = \lambda^j$. I get $ \lambda^{j+2} - \lambda^{j+1} -\lambda^{j} = 0 $. I'm not interested in $ \lambda = 0 $, so I've consider $ \lambda^{2} - \lambda^{1} - 1 = 0 $. I get $ \lambda_1 = 0.5 + \sqrt(1.25) $ and $ \lambda_2 = 0.5 - \sqrt(1.25) $. So
$z_j = c_1 * (0.5 + \sqrt(1.25))^j + c_2 * (0.5 - \sqrt(1.25))^j$ .
Last step : with the help of $ z_0=2 $ and $ z_1 = 1 $ I've got $c_1$ and $c_2$.
Now the second difference equation: same method. I get $(\lambda -1 )^3 *( \lambda + 3 ) = 0 $. So $ \lambda_1 = 1$ (multiple zero) and $ \lambda_2 = 3 $.Here is my problem: I've got a multiple zero so setting $ z_j = c_1 * 1^j + c_2 * (-3)^j $ is obviously false. Can you help me out?