卢卡斯数列和斐波那契数列关系性质新解  

New Discussions on Connections between Lucas Sequence and Fibonacci Sequence

作　　者：李艳平[1] 马丽娜[1] 鲁来凤[1] LI Yanping;MA Lina;LU Laifeng(School of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710119, China)

机构地区：[1]陕西师范大学数学与信息科学学院

出　　处：《高等数学研究》2019年第5期44-47,共4页Studies in College Mathematics

基　　金：陕西省重点研发项目(2019GY-013);中央高校基本科研业务费(GK201503013,GK201903011,GK201803005)资助;陕西师范大学研究生教育教学改革研究项目(GERP-19-41)

摘　　要：本文探讨通项公式非常相似的斐波那契数列{Fn}和卢卡斯数列{Ln}之间新的关系、性质和变化趋势.发现任何一个卢卡斯数Ln均可表达成两个斐波那契数Fn+1,Fn-1之和,而两个卢卡斯数Ln+1,Ln-1之和却等于5Fn;在讨论{Fn}和{Ln}前后比值数列{an/an+1}趋近于黄金数时,发现{an/an+1}的奇偶子列具有严格单调性和有界性;最后给出下一步关于{Fn}和{Ln}的研究思路.Based on the similarity of Fibonacci sequence {Fn} and Lucas sequence {Ln}, this paper explores their new connections, properties and the changing trends. It is found that any Lucas number can be expressed as the sum of two Fibonacci numbers, and the subsequences {a2n-1/a2n} and {a2n/a2n+1} of the ratio sequences {an/an+1} of {Fn} and {Ln} are strictly monotonic and bounded and convergent to the golden number. Some applications of {Fn} and {Ln} are also given.

关 键 词：卢卡斯数列 斐波那契数列 单调性 

分 类 号：O122[理学—数学]