For what value of C does this pattern converge? If we have a series of terms that follows the pattern of $x_{i}=\frac{x_{i-1}+x_{i-2}}{Cx_{i-1}},$ the pattern eventually diverges towards two oscillating values, for most values of $C$. Given that this series should converge towards $2/C$ as $i$ tends towards $\infty$, there should be a value for $C$ that allows this series to converge, but I only have a rough estimate of it (approx. $0.551834$). Is there a way to find the exact value?
Edit: $x_1$ = 1, $x_2$ = 2 for my particular example

If you define $y_i=Cx_i$, then $y_i=1+\dfrac{y_{i-2}}{y_{i-1}
}$, so the iteration is independent of $C$, and the only effect of $C$ is to adjust the starting values. You are then trying to find the starting values which give convergence to the fixed point of the iteration. As you say, for most starting values, you will get convergence to the 2-cycle $y_{n}=a, y_{n+1}=b$, where $(a-1)(b-1)=1$ and you want the starting values with $y_2=2y_1$ giving $a=b=2$. I think this simplifies, but does not solve, the problem.
