Here's one way of solving via the generating function method.
[tex]\begin{cases}a_0=3\\a_1=6\\a_2=0\\a_n=2a_{n-1}+a_{n-2}-2a_{n-3}&\text{for }n\ge3\end{cases}[/tex]
For the sequence [tex]a_n[/tex], denote its generating function by [tex]G(x)[/tex] with
[tex]\displaystyle G(x)=\sum_{n\ge0}a_nx^n[/tex]
In the recurrence relation, multiply all terms by [tex]x^n[/tex] and sum over all non-negative integers larger than 2:
[tex]\displaystyle\sum_{n\ge3}a_nx^n=2\sum_{n\ge3}a_{n-1}x^n+\sum_{n\ge3}a_{n-2}x^n-2\sum_{n\ge3}a_{n-3}x^n[/tex]
The goal is to rewrite everything we can in terms of [tex]G(x)[/tex] and (possibly) its derivatives. For example, the term on the LHS can be rewritten by adding and subtracting the the first three terms of [tex]G(x)[/tex]:
[tex]\displaystyle\sum_{n\ge3}a_nx^n=\sum_{n\ge0}a_nx^n-(a_0+a_1x+a_2x^2)=G(x)-3-6x[/tex]
For the other terms on the RHS, you need to do some re-indexing of the sum:
[tex]\displaystyle\sum_{n\ge3}a_{n-1}x^n=\sum_{n\ge2}a_nx^{n+1}=x\sum_{n\ge2}a_nx^n=x\left(\sum_{n\ge0}a_nx^n-(a_0-a_1x)\right)=x\bigg(G(x)-3-6x\bigg)[/tex]
[tex]\displaystyle\sum_{n\ge3}a_{n-2}x^n=\sum_{n\ge1}a_nx^{n+2}=x^2\sum_{n\ge1}a_nx^n=x^2\left(\sum_{n\ge0}a_nx^n-a_0\right)=x^2\bigg(G(x)-3\bigg)[/tex]
[tex]\displaystyle\sum_{n\ge3}a_{n-3}x^n=\sum_{n\ge0}a_nx^{n+3}=x^3\sum_{n\ge0}a_nx^n=x^3G(x)[/tex]
So in terms of the generating function, the recurrence can be expressed as
[tex]G(x)-3-6x=2x\bigg(G(x)-3-6x\bigg)+x^2\bigg(G(x)-3\bigg)-2x^3G(x)[/tex]
[tex](1-2x-x^2+2x^3)G(x)=3-15x^2[/tex]
[tex]G(x)=\dfrac{3-15x^2}{1-2x-x^2+2x^3}=\dfrac{3-15x^2}{(1-x)(1+x)(1-2x)}[/tex]
Decomposing into partial fractions, we get
[tex]G(x)=\dfrac6{1-x}-\dfrac2{1+x}-\dfrac1{1-2x}[/tex]
and we recognize that for appropriate values of [tex]x[/tex], we can write these as geometric power series:
[tex]G(x)=\displaystyle6\sum_{n\ge0}x^n-2\sum_{n\ge0}(-x)^n-\sum_{n\ge0}(2x)^n[/tex]
Or, more compactly,
[tex]G(x)=\displaystyle\sum_{n\ge0}\bigg(6-2(-1)^n-2^n\bigg)x^n[/tex]
which suggests that the solution to the recurrence is
[tex]a_n=6-2(-1)^n-2^n[/tex]
