Discrete Green convolution for the period-six recurrence ($P_k={\sum }_{j=1}^k{a}_{k-j+1}{f}_j,a_n=\frac{2}{\sqrt{3}}\sin (n\pi /3)$)

The unit-response sequence $a_1=a_2=1$, $a_{n+2}-a_{n+1}+a_n=0$ is six-periodic. In the displayed convolution, all older terms cancel under $P_{k+2}-P_{k+1}+P_k$ by this recurrence; the two new boundary terms reduce to $f_{k+2}$. Thus it gives a particular solution of the inhomogeneous linear recurrence with forcing $f_{k+2}$. Add the two-dimensional homogeneous span of $\cos (k\pi /3)$ and $\sin (k\pi /3)$ for the general solution. This is a discrete Green function construction.