  We evaluate a curious determinant, first mentioned by George Andrews in 1980
in the context of descending plane partitions. Our strategy is to combine the
famous Desnanot-Jacobi-Dodgson identity with automated proof techniques. More
precisely, we follow the holonomic ansatz that was proposed by Doron Zeilberger
in 2007. We derive a compact and nice formula for Andrews's determinant, and
use it to solve a challenge problem that we posed in a previous paper. By
noting that Andrews's determinant is a special case of a two-parameter family
of determinants, we find closed forms for several one-parameter subfamilies.
The interest in these determinants arises because they count cyclically
symmetric rhombus tilings of a hexagon with several triangular holes inside.
