The armchair (n,n) single walled carbon nanotubes with n=2—20 are studied by using the first principle density functional theory at the B3LYP/3-21G(d) level of theory combined with the periodic boundary conditions in simulating the ultra long tube model. The structure parameter, the energy, the band structure, and the energy gaps are obtained. The results show that the tube diameter and the energy of formation are closely related to n. The fitted analytical equations are developed with a correlation coefficient larger than 0.999. The energy gaps of (2,2) and (3,3) carbon nanotubes are 1.836 eV and 0.228 eV and the tubes have indirect energy gaps. For n=4 to 20, the energy gaps are quite small (between 0.027 eV and 0.079 eV), showing metal conductivity as well as direct energy gaps.