On the Inhomogeneous Geometric Line-Sequence

Abstract

There have been some recent investigations in the inhomogeneity of the second order recurrence sequence, see for example [1], [2], [3], [6] and [7]. We extend the investigation to include the inhomogeneity also in the corresponding geometric sequence. Consider the line-sequence generated by the recurrence relation

$$u_n = cu_{n-2} + bu_{n-1} + k, \ \ n \in z$$

, where c and b are non-zero integers and k the linear inhomogeneous term. We seek the conditions under which the terms of the line-sequence generated by (1.1) also satisfy the inhomogeneous geometric relation

$$u_n = xu_{n-1} + g$$

, where x is the geometric ratio and g the geometric inhomogeneous term.