Differential Equation - Series - Recurrence Relation

[mirrorx]

1. (16+x²)-xy'+32y=0

Seek a power series solution for the given differential equation about the given point x₀ find the recurrence relation.
So I used y=∑A_nxⁿ , found y' and y''
then I substituted it into the original equation, distributed, made all x to the n power equal to xⁿ, made the indexes 0, and added them all up.

Then I solved for a_n+2 and got:

(a_n-32a_n-a_nn(n-1))/(16(n+2)(n+1))=a_n+2

The question asks for for the recurrence relation in the form of a_2k+2 and a_2k+3
which are supposed to be the recurrence relation for even and odd terms.

How do I put it into that format? I'm just not sure where to go from this point. Also can someone even verify if I did the first part of obtaining a_n+2 correctly?

 

[mirrorx]

Picture of attempted solution

 

[mirrorx]

Never mind, I figured out the question. You can just substitute 2n+1 and 2n into all n's to get odd and even terms respectively.