I am looking for a possible proof of a formula.
I have been trying working on matrices and observed a pattern for the trace of positive integer power of the following matrix- $$A = \begin{bmatrix}0&0&a\\0&b&0\\c&0&0\end{bmatrix}$$ which is $$Tr(A^n) = \begin{cases} a^n, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text{$n$ odd} \\ a^n+2b^{n/2} c^{n/2},\,\,\,\,\, \text{$n$ even}\\ \end{cases}$$
What could possibly be the proof of this formula other than the iterative approach?
(i.e. if we try proving this formula as a theorem)
Please suggest some proof. Thankyou in advance :)
Let$$P=\begin{bmatrix}0&-\sqrt a&\sqrt a\\1&0&0\\0&\sqrt c&\sqrt c\end{bmatrix};$$the columns of $P$ are eigenvectors of $A$ and$$P^{-1}.A.P=\begin{bmatrix}b&0&0\\0&-\sqrt a\sqrt c&0\\0&0&\sqrt a\sqrt c\end{bmatrix}.$$So,$$\operatorname{tr}(A^n)=\operatorname{tr}\left(\begin{bmatrix}b^n&0&0\\0&\left(-\sqrt a\sqrt c\right)^n&0\\0&0& \left(\sqrt a\sqrt c\right)^n\end{bmatrix}\right).$$
You can also use the fact that$$A^n=\begin{cases}\begin{bmatrix}a^{n/2}c^{n/2}&0&0\\0&b^n&0\\0&0&a^{n/2}c^{n/2}\end{bmatrix}&\text{ if $n$ is even}\\\begin{bmatrix}0&0&a^{(n+1)/2}c^{(n-1)/2}\\0&b^n&0\\a^{(n-1)/2}c^{(n+1)/2}&0&0\end{bmatrix}&\text{ otherwise,}\end{cases}$$a fact that can easily be proved by induction.