Recursion Recursion Recursion

Algebra Level 5

$\begin{eqnarray} x_{n+1}&=&2x_n+3y_n \\ y_{n+1}&=&x_n+2y_n \end{eqnarray}$ When solving the above system of recursions with initial terms of $$x_0 = 2, y_0 = 1$$, the formula for $$x_n$$ is equal to $\frac{(A-B\sqrt{C})^{n+1}+(A+B\sqrt{C})^{n+1}}{D}$ where $$A,$$ $$B,$$ $$C,$$ and $$D$$ are positive integers with $$\gcd(A,B,D)=1,$$ and $$C$$ is square-free. Find the value of $\sum_{k=B}^Ax_k+\sum_{k=D}^Cy_k$

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