= Discrete Green convolution for the period-six recurrence
{title2=$P_k=\sum_{j=1}^k a_{k-j+1}f_j,\quad a_n=\frac2{\sqrt3}\sin(n\pi/3)$}
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.
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