A Property of the Unit Digits of Recursive Sequences

Abstract

In this paper we show that a sequence {A _n} defined by the second order recurrence relation

$$A_{n+2} = u_{1}A_{n+1} + u_2A_n$$

, satisfies the congruence relation

$$A_{n+2k} \equiv u_{1}A_{n+k} + u_2A_n \text{ (mod 10)}$$

for all choices of integers u ₁ and u ₂ and all initial values A ₁ and A ₂ if and only if k = 1 or 5 (mod 24). In the rest of this section we summarize the relevant earlier work in this area.