斐波那契数列前$n$项的平方和

isee

即 记 Fibonacci 数列的通项为$f_n$:$f_1=f_2=1$，$f_{n+1}=f_{n}+f_{n-1}$，则求$$f_1^2+f_2^2+\cdots+f_n^2=?$$

其实是在某选则题中看到，判断$f_1^2+f_2^2+\cdots+f_{2019}=2f_{2018}f_{2020}$是否正确，发现这个是错的.

应该是斐波那契数列的一个性质，注意$f_n^2=f_n(f_{n+1}-f_{n-1})$，累和即得$$f_1^2+f_2^2+\cdots+f_n^2=f_nf_{n+1}.$$

青青子衿

回复 1# isee

\begin{align*}
\sum_{k=1}^{n}{\operatorname{Fib}\left(k\right)}&=\operatorname{Fib}(n+2)-1\\
\sum_{k=1}^{n}{\operatorname{Fib}\!\,^2\!\left(k\right)}&=\operatorname{Fib}(n)\operatorname{Fib}(n+1)\\
\sum_{k=1}^{n}{\operatorname{Fib}\!\,^3\!\left(k\right)}&=\dfrac{\operatorname{Fib}(3n+2)+6(-1)^{n+1}\operatorname{Fib}(n-1)+5}{10}\\
\sum_{k=1}^{n}{\operatorname{Fib}\!\,^4\!\left(k\right)}
&=\dfrac{\operatorname{Fib}(4n+2) + 4(-1)^{n+1}\operatorname{Fib}(2n+1)+6n+3}{25}\\
\end{align*}

9.5 Summations of order > 2
\begin{align*}
\sum_{k=1}^{n}{\operatorname{Fib}\!\,^3\!\left(k\right)}
&=\dfrac{\operatorname{Fib}(n)\operatorname{Fib}\!\,^2\!(n+1)+(-1)^{n-1}\operatorname{Fib}(n-1)+1}{2}\\
\end{align*}

\begin{align*}
&F_{n}\color{red}{-}\phantom{1}F_{n-1}\color{red}{-}\phantom{01}F_{n-2}=0\\
&F_{n}^2\color{red}{-2}F_{n-1}^2\color{red}{-\phantom{0}2}F_{n-2}^2\color{orange}{+\phantom{01}}F_{n-3}^2=0\\
&F_{n}^3\color{red}{-3}F_{n-1}^3\color{red}{-\phantom{0}6}F_{n-2}^3\color{orange}{+\phantom{0}3}F_{n-3}^3\color{orange}{+\phantom{01}}F_{n-4}^3=0\\
&F_{n}^4\color{red}{-5}F_{n-1}^4\color{red}{-15}F_{n-2}^4\color{orange}{+15}F_{n-3}^4\color{orange}{+\phantom{0}5}F_{n-4}^4\color{green}{-\phantom{0}}F_{n-5}^4=0\\
&F_{n}^5\color{red}{-8}F_{n-1}^5\color{red}{-40}F_{n-2}^5\color{orange}{+60}F_{n-3}^5\color{orange}{+40}F_{n-4}^5\color{green}{-8}F_{n-5}^5\color{green}{-}F_{n-6}^5=0
\end{align*}
\begin{array}{|r|c|l|}
\hline {1}&\color{red}{-1}&\color{red}{-1}\\
\hline {1}&\color{red}{-2}&\color{red}{-2}&\color{orange}{+1}\\
\hline {1}&\color{red}{-3}&\color{red}{-6}&\color{orange}{+3}&\color{orange}{+1}\\
\hline {1}&\color{red}{-5}&\color{red}{-15}&\color{orange}{+15}&\color{orange}{+5}&\color{green}{-1}\\
\hline {1}&\color{red}{-8}&\color{red}{-40}&\color{orange}{+60}&\color{orange}{+40}&\color{green}{-8}&\color{green}{-1}\\
\hline \end{array}