A graph is pancyclic if it contains cycles of each length , . The generalized bull is obtained by associating one endpoint of each of the paths and with distinct vertices of a triangle. Gould, Łuczak and Pfender (2004) [4] showed that if is a 3-connected -free graph with then is pancyclic. In this paper, we prove that every 4-connected, claw-free, -free graph with is pancyclic. As the line graph of the Petersen graph is -free for any and is not pancyclic, this result is best possible.