the general solution of $y(n+3)-\frac{2}{3}y(n+1)+\frac{1}{3}y(n) = 0$ I have some trouble finding the correct solution for the difference equation
$$y(n+3)-\frac{2}{3}y(n+1)+\frac{1}{3}y(n) = 0$$
I've found that the characteristic equation of the difference equation is $\lambda^3-\frac{2}{3}\lambda+\frac{1}{3}$.
By computation I then have;
\begin{array}{lcl}
\lambda^3-\dfrac{2}{3}\lambda+\dfrac{1}{3}\\
(\lambda+1)(\lambda^2-\lambda+\dfrac{1}{3})\\
\lambda_1 = -1, \lambda_2 = \dfrac{1-i\sqrt{\frac{1}{3}}}{2}, \lambda_3 = \dfrac{1+i\sqrt{\frac{1}{3}}}{2}
\end{array}
I checked my answer using wolframalpha and it gave me the eigenvalues
$$\begin{array}{lcl}
\lambda_1 = -1, \lambda_2 = \dfrac{1-i\sqrt{3}}{2}, \lambda_3 = \dfrac{1+i\sqrt{3}}{2}
\end{array}$$
Could someone please tell me what I did wrong, or what I should do differently?

Wolfram Alpha gives the roots as
$-1$ and
$1/6 (3\pm i \sqrt(3))$.
This last is
$\frac{1\pm i\sqrt{1/3}}{2}$
which is your answer.
My guess is
either you misread WA's answer
or you entered the equation incorrectly.
