Both layered multiple-anion compounds and homologous series are of interest for their electronic properties, including the ability to tune the properties by changing the nature or number of the layers. Here we expand, using both computational and experimental techniques, a recently reported three-anion material, Bi4O4Cu1.7Se2.7Cl0.3, to the homologous series Bi2+2nO2+2nCu2-delta Se2+n-delta X delta (X = Cl, Br), composed of parent blocks that are well-studied thermoelectric materials. All of the materials show exceptionally low thermal conductivity (0.2 W/mK and lower) parallel to the axis of pressing of the pellets, as well as narrow band gaps (as low as 0.28 eV). Changing the number of layers affects the band gap, thermal conductivity, carrier type, and presence of a phase transition. Furthermore, the way in which the different numbers of layers are accessed, by tuning the compensating Cu vacancy concentration and halide substitution, represents a novel route to homologous series. This homologous series shows tunable properties, and the route explored here could be used to build new homologous series out of known structural blocks.