A non-automatic (!) application of Gosper's algorithm evaluates a   determinant from tiling enumeration
  We evaluate the determinant $\det_{1\leq i,j\leq n}(\binom{x+y+j}{x-i+2j}-\binom{x+y+j}{x+i+2j})$, which gives the number of lozenge tilings of a hexagon with cut off corners. A particularly interesting feature of this evaluation is that it requires the proof of a certain hypergeometric identity which we accomplish by using Gosper's algorithm in a non-automatic fashion. 
Mihai Ciucu (Georgia Institute of Technology, Atlanta) and Christian   Krattenthaler (Universit\"at Wien)