关于Lucas数列奇偶数项平方的倒数和公式  

On the Sum of Squares of Odd Even Terms of Lucas Sequence

作　　者：陈小芳 CHEN Xiaofang(School of Mathematics and Physics,Weinan Normal University,Weinan Shaanxi 714099,China)

机构地区：[1]渭南师范学院数理学院,陕西渭南714099

出　　处：《西华师范大学学报（自然科学版）》2017年第4期410-414,共5页Journal of China West Normal University(Natural Sciences)

基　　金：国家自然科学基金(11501419);渭南师范学院军民融合项目(16JMR11);渭南师范学院科研项目(17YKS11)

摘　　要：Fibonacci数列与Lucas数列均是著名数列,两者之间有着许多相同或类似的性质,且有着不少的联系。基于国外太多文献讨论Fibonacci数列的倒数的无穷和公式,Fibonacci数列偶数项平方的倒数的无穷和公式,奇数项平方的倒数的无穷和公式以及Fibonacci数列连续两项的乘积的倒数无穷和等,根据Lucas数列的定义和性质,讨论了Lucas数列奇数项平方的倒数和与Lucas数列偶数项平方的倒数和,结合数论相关知识,得出了结论并运用初等方法和解析方法给出了证明,具有一定的理论意义。As the famous series,the Fibonacci numbers and the Lucas numbers share the same or similar properties and they are closely related to each other.Many foreign literatures and documents have discussed about the Fibonacci sequence,including the infinite formula of the reciprocal about the Fibonacci sequence,the infinite sum formula of the reciprocal on the square of the even terms of the Fibonacci sequence,the infinite sum formula of the reciprocal on the square of the odd terms of the Fibonacci sequence,the reciprocal sum formula of the two consecutive terms product about the Fibonacci sequence and so on.based on these previous studies,the infinite sum formula of the reciprocal on the square of the odd terms of Lucas sequence and the infinite sum formula of the reciprocal on the square of the even terms of Lucas sequence are discussed in accordance with the definition and property of Lucas sequence.In combination with the number theory,conclusions are made and proved by the elementary method and analytic method,which is of certain theoretical significance.

关 键 词：LUCAS数列 倒数 无穷和 平方 奇数 偶数 

分 类 号：O156.2[理学—数学]