I've been trying to solve the following equation
$$x_{n+3}-x_n=2n+3\sin\frac{2n\pi}{3},\ \ n=0,1,\ldots$$ I am looking for my solution in the form of $$x_n=x_n^H+x_n^P$$, that is, as the sum of the homogoneous and particular solution. The homogenous is pretty easy, i have the characteristic equation $\lambda^3-1=0$, from where i get that $$\lambda_1=\lambda_2=\lambda_3=1,$$ and the fundamental set is $$\{1^n,n\cdot 1^n, n^2\cdot 1^n\},$$ and the homogeneous solution is $$x_n^H=c_1+nc_2+n^2c_3.$$ The problem is the particular solution. My question is, do i look for the particualr equation in the following form $$x_n^p=(An+B)\cdot n^3 + C\sin\frac{2n\pi}{3}+D\cos\frac{2n\pi}{3}?$$