When solving a recurrence relation with multiple repeated roots, would I add the two general solutions I get from each root?
For example this recurrence relation:
$a_{n+5} + 7a_{n+4} + 19a_{n+2} + 16a_{n+1} + 4a_{n} = 0$
Gives me the characteristic roots of $(r+1)^3(r+2)^2$, which is $r = -1$ with a multiplicity of $3$ and $r = -2$ with a multiplicity of $2$.
The general solutions are then $a_n=(\alpha_0 + \alpha_1n + \alpha_2n^2)(-1)^n$ and $a_n = (\alpha_0 + \alpha_1n)(-2)^n$, respectively. Would I then add the two right hand sides of the solutions to get the general solution of the entire recurrence relation?