We study a new class of networks, generated by sequences of letters taken from a finite alphabet consisting of $m$ letters (corresponding to $m$ types of nodes) and a fixed set of connectivity rules. Recently, it was shown how a binary alphabet might generate threshold nets in a similar fashion [Hagberg et al., Phys. Rev. E 74, 056116 (2006)]. Just like threshold nets, sequence nets in general possess a modular structure reminiscent of everyday life nets, and are easy to handle analytically (i.e., calculate degree distribution, shortest paths, betweenness centrality, etc.). Exploiting symmetry, we make a full classification of two- and three-letter sequence nets, discovering two new classes of two-letter sequence nets. The new sequence nets retain many of the desirable analytical properties of threshold nets while yielding richer possibilities for the modeling of everyday life complex networks more faithfully.