Four-term leaping recurrence relations

作者：Komatsu Takao^*

来源：Linear and Multilinear Algebra, 2010, 58(7): 875-886.

DOI：10.1080/03081080903125415

• leaping recurrence relations
• leaping convergents
• four-term recurrence relations
• Tribonacci numbers
• continued fractions

Consider linear three-term recurrence relation Zn = a1Zn-1 + a2Zn-2 (n epsilon 2). Then, from our former theorem, the three-term leaping recurrence relation [image omitted] holds for any positive integer k. In this article we shall consider the linear four-term recurrence relation Zn = a1Zn-1 + a2Zn-2 + a3Zn-3 (n epsilon 3). Then we can have the four-term leaping recurrence relation Zn = b1Zn-k + b2Zn-2k + b3Zn-3k for a given positive integer k with 1 k n/3.