Dynamical System , Series : can't find the general terms

Question

I have a dynamical system defined as follow :

$$V_{n+3} - 6V_{n+2} +12V_{n+1} - 8V_n = 8, ~ \mbox{with}~ V_0=V_1=V_2=1$$

I have to find $V_n$ = ?

So I began by solving this equation :

$$x^3 -6x^2 + 12x - 8 = 0$ <=> $(x-2)^3 = 0$ <=> $x= 2$$

But then I'd like to do :

$$V_n = \sum(\alpha_i*x_i^n))$$

but with $V_0 =V_1=V_2 = 1$ it seems kinda impossible ... Do you have an idea?

Answer 1

Mathematica's FindSequenceFunction states that $$V_n = \frac{1}{16} \left(9\ 2^n n^2-63\ 2^n n+63\ 2^{n+1}-128\right).$$ Once you have the formula, of course, it's an easy process to verify that it satisfies the recurrence relation.

For information on the mathematics behind this sort of thing, you might have a look at generatingfunctionology.