We have performed a detailed density functional theory study on the structural and electronic properties of Na(n)C(60)(-) (n = 1-12) clusters. The calculated vertical detachment energies show good agreement with the experimental data, which confirms the 3p (n = 3p) oscillation rule. The oscillation can be attributed to the combination of the charge depletion distribution induced by removing electrons and the number of the sodium atoms in direct contact with the fullerene. Based on the structural and electronic properties, the Na atoms can be categorized into two groups, one is for the metal atoms directly bonded to the fullerene surface, and the other one is for those without bonding to the fullerene. The Na atoms in group one would donate electrons to both the fullerene and the Na atoms in group two. As the total number of the sodium atoms increases, the number of Na atoms in group one would continue increasing till the size n = 3p - 1 to meet a shoulder from n = 3p - 1 to n = 3p, which accounts for the maximum vertical detachment energy at the size of n = 3p as drawn from the detailed electronic property studies.