The lattice cluster theory (LCT) is used to describe the monomer sequence dependence of the averaged free energy for an A-co-B random copolymer melt. This description is possible because the LCT considers local correlations (i.e., nonrandom mixing) and allows chemically different monomers to have different sizes and shapes as well as different interaction energies. In contrast tr,earlier theoretical approaches, the LCT treatment requires no ad hoc assumptions concerning the particular form of the sequence-dependent interaction parameters and employs no new adjustable parameters beyond those already appearing in the LCT far A/B homopolymer blends and for A-b-B diblock copolymer melts. The sequence-dependent contributions are extracted by comparing individual LCT diagrams for the free energy of random and diblock copolymer melts. Terms that differ for these two systems are identified as monomer sequence dependent. The sequence-dependent contributions are separated into enthalpic and entropic components. A detailed analysis of the local monomer topologies contributing to these diagrams demonstrates that the leading order sequence-dependent terms in the averaged free energy of random copolymer melts (as well as more complex mixtures) arise from diads of sequential monomers, not from triads as previously assumed.