We investigate how the coupled cluster method at the level of doubles and triples amplitudes contributes to the ground state energy of the homogeneous electron gas. We present and derive the formalism and equations needed, and describe in detail how two independent and conceptually differing computational schemes may be implemented efficiently for the system under study. We finally perform numerous calculations for the infinite electron gas, investigate how the gradual inclusion of more diagrams leading up to the full coupled cluster doubles triples (CCDT) method affects the energy, and we estimate the energy in the thermodynamical limit by extrapolating results from large scale computations. In order to check all equations, we have also developed a software which produces all equations needed at a given level of truncation of coupled cluster theory. This allows for efficient benchmarking of equations as well as codes for implementing various conributions to the theory.