I solved this linear difference equation $$x_{k+2} = c_1 x_{k+1} + c_2 x_k$$ to get the general solution
\begin{align} x_k = \lambda \left(\frac{c_1 + \sqrt{c_1^2 + 4 c_2}}{2} \right)^k + \mu \left(\frac{c_1 - \sqrt{c_1^2 + 4 c_2}}{2} \right)^k \end{align}
The roots are $$R_{\pm} = \frac{c_1 \pm \sqrt{c_1^2 + 4 c_2}}{2}$$
What if $R_{\pm}$ are complex? What is the interpretation of that?
$x_k$ should be real because $c_1, c_2 \in \mathbb{R}$ so how to write the solution in this case?
I recall something like writing a linear combination of real and imaginary parts of the roots in this case as follows:
$$x_k = \lambda \left ( \text{Re}(R_+) \right )^k + \mu \left ( \text{Im}(R_+) \right )^k $$
If this is true, why do we do that?
The general solution will also be $$ x_k = \lambda R_{+}^k + \mu R_{-}^k $$ but it requires some effort to see why this is real and how we can modify the expression in such way that we do not have to deal with complex numbers.
$R_{+}$ and $R_{-}$ are the roots of the polynomial $t^2-c_1t-c_2.$ Complex roots of real polynomials form pairs of conjugate complex numbers, this means $R_{-}=\overline R_{+}.$
Let us write $R_{+}$ and $R_{-}$ in polar form, $R_{+} = re^{\varphi i}$ and $R_{-} = re^{-\varphi i}.$ Furthermore, we want $$ \lambda\phantom{R_{+}} + \mu\phantom{R_{-}} = x_0 \in \mathbb{R} \\ \lambda R_{+} + \mu R_{-} = x_1 \in \mathbb{R} $$ If we solve this system of equations for $\lambda$ and $\mu,$ then we notice that $\lambda$ and $\mu$ also form a pair of conjugate complex numbers, $\mu = \overline \lambda.$ So let us write them in polar form, too: $\lambda = se^{\theta i}$ and $\mu = se^{-\theta i}$
Put all this together: $$ x_k = \lambda R_{+}^k + \mu R_{-}^k =se^{\theta i}r^k e^{k\varphi i} + se^{-\theta i}r^k e^{-k\varphi i} =sr^k\left(e^{(k\varphi +\theta)i}+e^{-(k\varphi +\theta)i} \right) \\ =2sr^k\cos(k\varphi +\theta) $$ and we got rid of all complex numbers.
As you can see, the sequence has an oscillating behavior. $\varphi$ determines the period, $\theta$ the phase, $s$ determines the initial amplitude, and $r$ determines whether the absolute values of the numbers become larger or smaller.