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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.

An inhomogeneous linear recurrence has a prescribed forcing sequence $g_n$ rather than zero on the right. The difference of two solutions satisfies the associated homogeneous linear recurrence relation, so every solution is a particular solution plus a homogeneous solution.

The forcing base $-2$ is a root of the characteristic equation of a linear recurrence, so a constant multiple of $(-2{)}^n$ cannot be a particular solution. For $P(r)=r^3-3r+2$, direct substitution givesFor the resonant case of exponential forcing in a linear recurrence, at $r=-2$, $P(-2)=0$ and $(-2)P^{\prime }(-2)=-18$. Hence $-(n/18)(-2{)}^n$ is a particular solution of the inhomogeneous linear recurrence. The general solution is

Let $L[y{]}_n=y_{n+3}-3y_{n+1}+2y_n$. For the exponential trial sequence $D{2}^n$,Thus $D=1/4$ gives a particular solution. The difference of any two solutions of the inhomogeneous linear recurrence satisfies the homogeneous linear recurrence relation in part (i). Therefore

Eliminate the free constant by comparing consecutive values:Therefore an appropriate inhomogeneous linear recurrence isIts homogeneous part has solution $a{2}^n$, and $-n$ is a particular solution. A first-order linear recurrence relation has one arbitrary initial value, so this gives exactly the desired general solution.