The combinatorial objects discussed in this paper, polyominoes, are connected finite unions of unit squares in the plane. Among the parameters of interest in enumerating polyominoes are area (number of squares) and perimeter (length of the border). In attempting to obtain a formula that can enumerate a class of objects, such as the number of convex polyominoes with a given perimeter, the need arises for handling large algebraic expressions. For this purpose, the MACSYMA system (and, more recently, the MAPLE system) was used. This paper reviews enumerative results obtained in previous papers by the author and his collaborators. The main discussion is devoted to the organization of the MACSYMA computations that were carried out and the requirements of a computer algebra system when used on such problems.