sequences and series - How to find the general formula for this recursive problem?

I want to ask about recursive problem.

Given:

$$a_0= 11, a_1= -13,$$ and $$a_n= -a_{n-1} +2a_{n-2}.$$

What is the general formula for

$$a_n$$ ?

I've already tried to find the first terms of this series. From there, I got:

$$a_2 = 35, a_3= -61,$$ and $$a_4= 131.$$

From there, I think I need to use the rule from arithmetic and geometric series to find the general formula that I want to find.

But, I cannot find the certain pattern from this series, because the differences is always changing, such that -24, 48, -96, 192.

From there I think the general formula for a_n should be including (-1)ⁿ. But, how can we deal with series 24,48,96,192?

It seems the series is geometric, but how can we find the formula?

Thanks

Answer

Linear Recurrence Equations have typical solutions $a_n=\lambda^n$. Using this, we can compute the possible values of $\lambda$ for this equation from

$$\lambda^n=-\lambda^{n-1}+2\lambda^{n-2}$$
which means, assuming $\lambda\ne0$, that
$$\lambda^2+\lambda-2=0$$
This is the characteristic polynomial for the recurrence
$$a_n=-a_{n-2}+2a_{n-2}$$

The characteristic polynomial is $x^2+x-2$ which has roots $1$ and $-2$. Thus, the sequence is $a_n=b(1)^n+c(-2)^n$. Plugging in the values for $n=0$ and $n=1$ gives

$$a_n=3+8(-2)^n$$