Hi all,

I am trying to slove the recursion equation: $x_{n+1}x_{n−1}=x_n^2(1−4x_n)$ in the form of $x_n=x_n(x_1,x_2)$ or $x_n=x_n(c_1,c_2)$, and finally get the limit of the ratio: $\dfrac{x_n}{x_{n+1}}$.

I tried the way of setting: $x_n=f(n)$, and use the 1st order taylor expansion of $f(n+1)$ and $f(n−1)$, then get the first differential equation: $(f')2=4f^3$, the general solution is: $f(x)=\dfrac{4}{(2x−c_1)^2}$, but it does not fit the original recursion equation.

Then I write the equation in the form: $f''=6f^2$, the computer provides some approach by Weierstrass elliptic function (http://www.wolframalpha.com/input/?i=d^2y%2Fdx^2%3D6y^2), but it seems the Weierstrass elliptic function still has no such property as the recursion formula.

Any method I cam apply to get the final limit of ratio, maybe without solving the general soltuions? Thanks!