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? 

Here’s a rote rule for doing so. Start with the recurrence:
$$U_n=3U_{n-1}-U_{n-3}$$
Convert each subscript to an exponent:
$$U^n=3U^{n-1}-U^{n-3}$$
Change the variable to the one that you want to use in the characteristic equation:
$$x^n=3x^{n-1}-x^{n-3}$$
Divide through by the smallest exponent, in this case $n-3$:
$$x^{n-(n-3)}=3x^{(n-1)-(n-3)}-1\;,$$
which simplifies to $$x^3=3x^2-1\;.$$
With a little practice you can do the conversion in one go. For instance, the recurrence $$a_n=4a_{n-2}-6a_{n-3}+3a_{n-4}$$ has characteristic equation $$x^4=4x^2-6x+3\;,$$ as you can check by following through the steps given above.
