About continued fractions expansions of metallic means

Abstract

It is well known that the simple continued fraction expansion of the Metallic Means ?p;1; are periodic. We begin obtaining new better generalized periodic expansions for this Metallic subfamily. The odd powers of the Metallic Means have continued fraction expansions in terms of the certain generalized Lucas numbers. Each of this power is also some Metallic Mean. We prove that the even powers of the Metallic Means, are always the solution of a quadratic equation x² - mx + 1 = 0, where the parameter m is also defined by means of another generalized Lucas numbers, and from this result, directly we achieve the generalized continued fraction expansion of this even powers.