How to get the characteristic equation? In my book, this succession defined by recurrence is presented:
$$U_n=3U_{n-1}-U_{n-3}$$
And it says that the characteristic equation of such is:
$$x^3=3x^2-1$$
Honestly, I don't understand how. How do I get the characteristic equation given a succession? 

Given a recurrence, $$a_{n+j+1} = \sum_{k=0}^{j} c_k a_{n+k}$$
Take $a_n = x^n$. Then the characteristic equation is $$x^{n+j+1} = \sum_{k=0}^{j} c_k x^{n+k}$$
which gives us the characteristic equation $$x^{j+1} - \sum_{k=0}^{j} c_k x^k = 0$$
This is analogous to taking $y = e^{mx}$ when we solve linear differential equations.
