关于a_(n+2)=pa_(n+1)+qa_n的周期性

• 作 者: 黄克之 黄军华
• 机 构: 深圳学而思,深圳市深圳中学,
• 出 处: 数学通讯
• 发表日期: 2013年18期
• 关键词: 通项公式 实数根 递推数列 n+1 n+2 pa qa_n 常数数列 特征方程 共轭虚根

在文[1]中,作者给出了二阶递推数列an+2=pan+1+qan(其中p,q均为常数)的特征方程y2-py-q=0满足如下条件时,其对应的通项公式如下:(1)若方程y2-py-q=0有两个不等实数根y1,y2时,通项公式为an=λ1yn1+λ2yn2;(2)若方程y2-py-q=0有两个相等实数根y0时,通项公式为an=(λ1+nλ2)yn0;(3)若方程y2-py-q=0有两个共轭虚根时,通项公式为an=rn(λ1cosnθ+λ2sinnθ);其中λ1,λ2由初始条件解出.

In [1], the author gives the characteristic equation y2-py-q = 0 for the second-order recursive sequence an + 2 = pan + 1 + qan where p and q are all constant. The corresponding general formula is as follows: (1) If the equation y2-py-q = 0 has two unequal real roots y1, y2, the general formula is an = λ1yn1 + λ2yn2; (2) (3) If the equation y2-py-q = 0 has two conjugate imaginary roots, the general formula is an = rn (λ1cosnθ + λ2sinnθ); where λ1, λ2 are solved by the initial conditions.