广义Fibonacci数列和Lucsa数列的关系式

张福玲

doi:10.3969/j.issn.1673-159X.2016.05.015

广义Fibonacci数列和Lucsa数列的关系式

张福玲

Some Relations of Generalized Fibonacci Sequence and Lucas Sequence

ZHANG Fuling

摘要

摘要: 根据广义的Fibonacci数列u_n:u_n+1=Au_n+Bu_n-1和广义Lucas数列v_n:v_n+1=Av_n+Bv_n-1的定义, 采用初等方法证明了广义的Fibonacci数列和Lucas数列的几个新的关系式\sum\limits_i = 0^n u_iv_n - i = \left( n + 1 \right)u_n 、 2^n + 1u_n + 1=\sum\limits_i = 0^n 2^iv_iA^n - i、 \sum\limits_i = 0^n \left( - B \right)^iv_n - 2i = 2u_n + 1 、 3^n + 1u_n + 1 = \sum\limits_i = 0^n 3^iv_iA^n - i + \sum\limits_i = 0^n + 1 3^i - 1u_iA^n + 1 - i 、 \sum\limits_i = 0^n v_iv_n - i = \left( n + 1 \right)v_n + 2u_n + 1 = \left( n + 2 \right)v_n + Au_n、 \left( A^2 + 4B \right)\sum\limits_i = 0^n u_iu_n - i = \left( n + 1 \right)v_n - 2u_n + 1 = nv_n - Au_n , 将Fibonacci数列和Lucsa数列关系的结论进行了推广。

Abstract: According to the definitions of generalized Fibonacci sequence and Lucas sequence, some relations of generalized Fibonacci and Lucas sequence were proved by using elementary method, such as, \sum\limits_i = 0^n u_iv_n - i = \left( n + 1 \right)u_n , 2^n + 1u_n + 1=\sum\limits_i = 0^n 2^iv_iA^n - i, \sum\limits_i = 0^n \left( - B \right)^iv_n - 2i = 2u_n + 1 , 3^n + 1u_n + 1 = \sum\limits_i = 0^n 3^iv_iA^n - i + \sum\limits_i = 0^n + 1 3^i - 1u_iA^n + 1 - i , \sum\limits_i = 0^n v_iv_n - i = \left( n + 1 \right)v_n + 2u_n + 1 = \left( n + 2 \right)v_n + Au_n, \left( A^2 + 4B \right)\sum\limits_i = 0^n u_iu_n - i = \left( n + 1 \right)v_n - 2u_n + 1 = nv_n - Au_n , and the relations of Fibonacci-Lucas sequences were promoted.

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