  We demonstrate a new kind of companion matrix, for polynomials of the form
$c(\lambda) = \lambda a(\lambda)b(\lambda) + c_0$ where upper Hessenberg
companions are known for the polynomials $a(\lambda)$ and $b(\lambda)$. This
construction can generate companion matrices with smaller entries than the
Fiedler or Frobenius forms. This generalizes Piers Lawrence's Mandelbrot
companion matrix. We motivate the construction by use of Narayana-Mandelbrot
polynomials, which are also new to this paper.
