Properties of a Certain Lyness Equation

We investigate oscillation, cycle length, and extreme values for the difference equation xnC1 D aCPk−1 iD0 bixn−i‘=xn−k, where a and bi are nonnegative numbers and a C Pk−1 iD0 bi‘ > 0. If a > 0, it is known from Theorem 2.2.1 of V. L. Kocic, G. Ladas, and I. W. Rodrigues (J. Math. Anal. Appl. 173, 1993, 127–157) that every cycle has no more than 2kC1‘ terms. If a D 0, we show this is not necessarily true. A general statement concerning cycle length if a D 0 remains an open question