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I've found a couple of interesting formulae for the characteristic polynomial $c_A(x)$ of matrices in the form of $$A= \begin{pmatrix} 0 & 1 & 0 & 0 & \cdots & 0 \\ 0 & 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & 0 & \cdots & 1 \\ 1 & 0 & 0 & 0 & \cdots & 0 \end{pmatrix} $$

That is $$c_A(x)=(-1)^n(x^n-1)$$

Are there any similar formulae for some special kinds of matrices for quick reference?

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    Google "companion matrix". What you wrote is precisely the companion matrix of the polynomial $\;\pm(x^n-1)\;$ ... – 2017-02-08
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    See also [Circulant matrix](https://en.wikipedia.org/wiki/Circulant_matrix#Eigenvectors_and_eigenvalues). – 2017-02-08

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It seems that you are looking at a special case of a companion matrix. Have a look here: Wikipedia.