Complex equation simplification Let $k$ be a positive integer and $c_0$ a positive constant. Consider the following expression:
\begin{equation} \left(2 i c_0-i+\sqrt{3}\right)^2 \left(-2 c_0+i \sqrt{3}+1\right)^k+\left(-2 i c_0+i+\sqrt{3}\right)^2 \left(-2 c_0-i \sqrt{3}+1\right)^k
\end{equation}
I would like to find a simple expression for the above in which only real numbers appear. It is clear that the above expression is always a real number  since
\begin{equation}
\overline{\left(2 i c_0-i+\sqrt{3}\right)^2 \left(-2 c_0+i \sqrt{3}+1\right)^k}=  \left(-2 i c_0+i+\sqrt{3}\right)^2 \left(-2 c_0-i \sqrt{3}+1\right)^k.
\end{equation}
But I am not able to simplify it.  I am pretty sure I once saw how to do this in a complex analysis course but I cannot recall the necessary tools. Help is much appreciated.
 A: i will outline one possible line of approach to this question. First, simplify a little. let the value of your expression be $E_k$ and let
$$
a = \frac{1-2c_0}{\sqrt{3}}
$$
then we may write
$$
\frac{E_k}{3^{\frac{k}2 +1}} = (1-ai)^2(a+i)^k + (1+ai)^2(a-i)^k
$$
using your remark concerning conjugacy we may write this as
$$
\frac{E_k}{3^{\frac{k}2 +1}} = 2\mathfrak{Re}\bigg((1-ai)^2(a+i)^k \bigg)
$$
now
$$
\mathfrak{Re}\bigg((1-ai)^2(a+i)^k \bigg) = \mathfrak{Re}(1-ai)^2 \mathfrak{Re}(a+i)^k - \mathfrak{Im}(1-ai)^2 \mathfrak{Im}(a+i)^k \\
= (1-ai)^2 \mathfrak{Re}(a+i)^k - \mathfrak{Im}(1-ai)^2 \mathfrak{Im}(a+i)^k \\
(1-a^2)\mathfrak{Re}(a+i)^k +2a\mathfrak{Im}(a+i)^k
$$
set 
$$
S_k = \mathfrak{Re}(a+i)^k \\
T_k = \mathfrak{Im}(a+i)^k
$$
so we have the recurrence relations:
$$
S_{k+1} = aS_k -T_k \\
T_{k+1} = S_k + a T_k
$$
since
$$
H_k =\mathfrak{Re}\bigg((1-ai)^2(a+i)^k \bigg) = (1-a^2)S_k +2aT_k
$$
and $S_0 =1$, $T_0 =0$
we may write
$$
H_k = \begin{pmatrix} 1-a^2 & 2a \end{pmatrix}\begin{pmatrix} a & -1 \\ 1 & a \end{pmatrix}^k\begin{pmatrix} 1 \\ 0 \end{pmatrix}
$$
given the unreliability of my arithmetic, i shall venture no further, as this formulation may contain errors. however the method outlined may be useful as a procedure for calculating the value for a particular $k$.
