Recent experiments have shown it is possible to produce striped surfaces with periodicities of the same order as those for self-organized diblock copolymer lamellae. Here we study theoretically the behavior of a thin film of AB diblocks on a CD striped surface. The competition between the preferred bulk lamellar spacing and the spacing favored by the stripes leads to several novel phases. We study this system by analytical and numerical means and in one limit by mapping it onto the Frenkel-Kontorowa model of solid state physics. The main conclusion is that the periodically striped surface can induce lamellae of unequal spacing. We also show that the surface can induce inverted bilayers where an AB is followed directly by another AB rather than a BA.