An infinite series of ͑3,6͒ cages is defined by trivalent carbon polyhedra composed of hexagonal and four triangular rings. A zone-folding construction is applied to the graphene band structure to yield explicit expressions for the -molecular orbitals, energies, and symmetries of the cages that depend only on four indices m, n, p, and q. Leapfrog members of the series ͑mϪnϭ0 mod 3 and pϪqϭ0 mod 3͒ have closed shells in a neutral form with two filled nonbonding orbitals; all others have closed shells as dications. Quantum chemical calculations on C 12 , C 48 , and C 52 2ϩ confirm this result. Embedding relationships are proved for the spectra of ͑3,6͒ cages related by inflation transformations corresponding to stretching and rotation of the polyhedral net.