We study the electronic states of giant single-shell and the recently discovered nested multi-shell carbon fullerenes within the tight-binding approximation. We use two different approaches, one based on iterations and the other on symmetry, to obtain the π-state energy spectra of large fullerene cages: C_240, C_540, C_960, C_1500, C_2160 and C_2940. Our iteration technique reduces the dimensionality of the problem by more than one order of magnitude (factors of ∼ 12 and 20), while the

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... based approach reduces it by a factor of 10. We also find formulae for the highest occupied and lowest unoccupied molecular orbital (HOMO and LUMO) energies of C_60·n^2 fullerenes as a function of n, demonstrating a tendency towards metallic regime for increasing n. For multi-shell fullerenes, we analytically obtain the eigenvalues of the intershell interaction.