Does this linear recurrence relation have a solution?

Question

It seems to me that this recurrence relation has no solution. The answer I get does not prove true for all of the initial conditions. I have also tried solving it from Wolfram Alpha, which says no solutions exist yet my marking scheme has an answer.

$$2a_{n+3}=a_{n+2}+2a_{n+1}-a_n$$ given that $a_0=0, a_1 = 1, a_2 = 2$

Answer in the marking scheme is:

$$a_{n}=5/2 - (1/6)^n - 8/3(1/2)^n$$

[Edit: I found my mistake; see my answer below.]

Answer 1

Wolfram Alpha gives the correct answer and I realized I was not doing the question correctly so the answer can be found using: $$a_n = q^n$$

The correct answer is: $$\frac{1}{6}[(-1)^n-2^{4-n}+15]$$

Answer 2

Your equation is linear, so try it with $$a_n=q^n$$ Then you will get the equation for $q$ $$2q^{n+3}=q^{n+2}+2q^{n+1}-q^n$$

Now compute $q$ Divide the whole equation by $q^n$