Cannot find the pattern in recurrence relation

[swtlilsoni]

Homework Statement

I am doing a power series solution for: (x^2-1)y" + 8xy' + 12y = 0
I rewrote it in terms of power series and transformed everything into one series and finally ended up with the following recurrence relation:
a_n+2= ((n+3)(n+4)a_n)/((n+2)(n+1))

I plugged in values for n but I am having a hard time finding a pattern. I need to turn this into a power series but I cannot find the pattern in order to do that. I am getting everything in terms of either a₀ or a₁ and I know that I need to group all the a₀'s together and that is one solution, and the a₁'s together and that is the second solution. However even after grouping them I cannot find the pattern.

 

[Sethric]

Without actually plugging in values for n, what does $$a_{n+4}$$ equal in terms of $$a_n$$? How about n+6? Figuring out the cancellation pattern should give you a formula a_n.