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Regularly coiled carbon nanotubes, their structure, and formation mechanism are puzzling questions. The first models were based on the very regular incorporation of a small fraction (of the order of 10%) of nonhexagonal (n-Hx) rings: (pentagons and heptagons) in a perfect hexagonal (Hx) lattice. It is difficult to understand by which mechanism takes place such a regular incorporation of isolated n-Hx rings. In this paper, a new family of Haeckelite nanotubes is generated in a systematic way by rolling up a two-dimensional three-fold coordinated carbon network composed of pentagon-heptagon pairs and hexagons in proportion 2 : 3. In this model, the n-Hx rings are treated like regular building blocks of the structure. Cohesion energy calculation shows that the stability of the generated three-dimensional Haeckelite structures falls between that of straight carbon nanotubes and that of C60. Electronic density of states of the Haeckelite computed with a tight-binding Hamiltonian that includes the C-π orbitals only shows that the structures are semiconductor. The relation of the structures with experimental observations is discussed.