  We report on the electronic structure, density of states and transmission
properties of the periodic one-dimensional Tight-Binding (TB) lattice with a
single orbital per site and nearest-neighbor interactions, with a generic unit
cell of $u$ sites. The determination of the eigenvalues is equivalent to the
diagonalization of a real tridiagonal symmetric $u$-Toeplitz matrix with
(cyclic boundaries) or without (fixed boundaries) perturbed upper right and
lower left corners. We solve the TB equations via the Transfer Matrix Method,
producing, analytical solutions and recursive relations for its eigenvalues,
closely related to the Chebyshev polynomials. We examine the density of states
and provide relevant analytical relations. We attach semi-infinite leads,
determine and discuss the transmission coefficient at zero bias and investigate
the peaks number and position, and the effect of the coupling strength and
asymmetry as well as of the lead properties on the transmission profiles. We
introduce a generic optimal coupling condition and demonstrate its physical
meaning.
