The possible ways of generation and growing the fullerenes having at first six-fold symmetry has been studied. Beginning with a hexagonal prism (elementary fullerene C12), six-cornered barrel-shaped fullerene C24 and high six-cornered barrel-shaped fullerene C36, we obtained their direct descendants throw the use of Endo-Kroto’s mechanism known as embedding carbon dimers. We have calculated the energies of the possible fullerenes and discussed possible reasons of their dependence on a fullerene size and shape.