I've recently taken an interest in the Method of Annihilators for solving recurrence relations. However, I taught myself using nearly solely this Power Point presentation. Thus, my knowledge is lacking in some ways.
My question is, how does one "come up" with an annihilator? The presentation I saw showed how to prove/demonstrate an annihilator works, but now how to just pull one out of thin air.
An example of what I'm talking about:
How would I find an annihilator for $\sin\left(\frac{\pi}{2}n\right)$? I could just start guessing/writing out the sequence, and stumble upon the annihilator $(E^2-1)$ But, is there a systematic way to find this annihilator?
The trick behind such terms is to consider them in the complex plane, I.e. $\sin \frac{\pi}{2} n = \frac{1}{2 \mathrm{i}} \left(\exp\left( \frac{\pi \mathrm{i}}{2}n\right) - \exp\left(-\frac{\pi\mathrm{i}}{2} n\right)\right)$, and this is just two $n$-th powers