In this paper we study the behaviour of the sequence of scalar ( real or complex) numbers satisfying the relation , where is a fixed sequence of scalar numbers. Such kind of sequences arise in problems of analysis, fixed point theory, dynamical sistems, theory of chaos and ets. For example, investigating the spectra of triple and more than triple band triangle operator-matrices arise above mentioned sequences which required to study the behaviour of these sequences. From the point of application, the proved results and formulas in the literature for the spectra of the operator-matrices look like very complicated. In this work for the eliminating of indicated flaws we apply new approach, where the formulas for the spectra for the generalized difference operator-matrices describe circular domains, and also we study problems describing the theory of natural processes.