On some generalizations of Horadam’s numbers

Abstract

In this paper, we introduce the incomplete Horadam numbers Wn(k), and hyper-Horadam numbers W(k)n, which generalize the Horadam?s numbers defined by the recurrence Wn = pWn-1 + qWn-2, with W0 = a and W1 = b. We give some combinatorial properties. As an application, we evaluate a lower and upper bounds for the spectral norms of r-circulant matrices associated with these two generalizations. Moreover, we establish a new bounds for the spectral norms of r-circulant matrices associated with Horadam?s numbers in terms of incomplete Horadam and hyper-Horadam numbers.