An α-p-catapoly-q-gon is a catacondensed polygonal system with α p-gons and r − α q-gons, which represents a polycyclic conjugated hydrocarbon. Here r is the total number of polygons or rings. A complete mathematical solution for the numbers of nonisomorphic unbranched α-p-catapoly-q-gons is reported. It is expressed by complicated explicit formulas in r, α, p, and q, and it represents a generalization of several special cases studied previously. The solution was achieved by means of certain generalized triangular matrices with interesting mathematical properties.