Four-term leaping recurrence relations

Abstract

Consider linear three-term recurrence relation Z n = a 1 Z n−1 + a 2 Z n−2 (n ≥ 2). Then, from our former theorem, the three-term leaping recurrence relation holds for any positive integer k. In this article we shall consider the linear four-term recurrence relation Z n = a 1 Z n−1 + a 2 Z n−2 + a 3 Z n−3 (n ≥ 3). Then we can have the four-term leaping recurrence relation Z n = b 1 Z n−k + b 2 Z n−2k + b 3 Z n−3k for a given positive integer k with 1 < k < n/3.