We use a tight-binding formulation to investigate the electronic density of states and the energy spectra of single and double-strand DNA molecules made up from the nucleotides guanine (G), adenine (A), cytosine (C), and thymine (T). A renormalization group approach is also employed to take into account the sugar–phosphate contribution. In order to reveal the relevance of the underlying correlations in the nucleotides distribution, we compare the results for the genomic DNA sequence with those of two artificial quasiperiodic sequences: the Fibonacci and Rudin-Shapiro ones, which have long-range correlations. In addition, we consider also a random sequence, which is a kind of prototype of a short-range correlated system, presented here with the same first-neighbor pair correlations of the human DNA sequence. We found that the long-range character of the correlations is important to the persistence of resonances of finite segments.