The real sequence $y_{k}, k=1,2, \ldots$ satisfies the difference equation

$$y_{k+2}-y_{k+1}+y_{k}=0$$

Show that the general solution can be written

$$y_{k}=a \cos \frac{\pi k}{3}+b \sin \frac{\pi k}{3},$$

where $a$ and $b$ are arbitrary real constants.

Now let $y_{k}$ satisfy

$$y_{k+2}-y_{k+1}+y_{k}=\frac{1}{k+2}$$

Show that a particular solution of $(*)$ can be written in the form

$$y_{k}=\sum_{n=1}^{k} \frac{a_{n}}{k-n+1},$$

where

$$a_{n+2}-a_{n+1}+a_{n}=0, \quad n \geq 1$$

and $a_{1}=1, a_{2}=1$.

Hence, find the general solution to $(*)$.