Let $k$ be a positive integer. Let
$$Q=
\begin{pmatrix}
 1 &1/2&   &   &   &   \\
1/2& 1 &   &   &   &   \\
   &   & 1 &1/2&   &   \\
   &   &1/2& 1 &   &   \\
   &   &   &   & 1 &1/2\\
   &   &   &   &1/2& 1 
\end{pmatrix}.
$$

> How many solution $x\in\mathbb Z^6$ are there to $\quad  x^tQx=k$?


This is equivalent to: 

> How many solution $x\in\mathbb Z^6$ are there to 
> $$x_1^2+x_1x_2+x_2^2+ x_3^2+x_3x_4+x_4^2+ x_5^2+x_5x_6+x_6^2=k?$$

or to

> How many solution $x\in \mathbb Z\left[\omega \right]^3$ are there to $\quad x^* I_3 x=k$?

where $I_3$ is the $3\times3$-identity matrix and $\omega=\frac{1+\sqrt{-3}}{2}$.

I know that there is a formula for this number (there is only one class in its genus), but I don't know it.

This question is related to

 - https://mathoverflow.net/questions/78361/which-integers-take-the-form-x2-xy-y2
 - https://math.stackexchange.com/questions/44139/how-many-solutions-are-there-to-fn-m-n2nmm2-q/

but they don't answer my question.