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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