k 次斐波那契数列的线性递推关系

葡萄糖

性质：\(k\) 次 \(Fibonacci\) 数列 \(\{F_n^k\}\) 中连续 \(k+2\) 个数的线性递推关系：

\begin{align*}F_{n}=-F_{n+1}+F_{n+2} &\iff F_{n}=F_{n-1}+F_{n-2}\\
F_{n}^2=2F_{n+1}^2+2F_{n+2}^2-F_{n+3}^2 &\iff F_{n}^2=2F_{n-1}^2+2F_{n-2}^2-F_{n-3}^2\\
F_{n}^3=-3F_{n+1}^3+6F_{n+2}^3+3F_{n+3}^3-F_{n+4}^3 &\iff F_{n}^3=3F_{n-1}^3+6F_{n-2}^3-3F_{n-3}^3-F_{n-4}^3\end{align*}

mathe

设x^2-x-1的根为w,-1/w.
k次特征方程乘上(x-w^k)(x-(-1/w)^k)=x^2-L_k+(-1)^k就变成k+2次特征方程，其中L_k和F_k有相同递推式

gxqcn

\(\hphantom{\iff} F_{n}^3=3F_{n-1}^3+6F_{n-2}^3-3F_{n-3}^3-F_{n-4}^3\)
\(\iff F_{n-4}^3=-3F_{n-3}^3+6F_{n-2}^3+3F_{n-1}^3-F_{n}^3\)
\(\iff F_{n}^3=-3F_{n+1}^3+6F_{n+2}^3+3F_{n+3}^3-F_{n+4}^3\)

所以，主题帖里，没有必要贴出“\(\iff\)”左边的部分。

更一般地，用方程式的形式更优雅：\(F_{n}^3-3F_{n-1}^3-6F_{n-2}^3+3F_{n-3}^3+F_{n-4}^3=0\)，
大家只需记住前面的系数即可：\((1,-3,-6,3,1)\)

葡萄糖

gxqcn 发表于 2015-6-25 11:04
\(\hphantom{\iff} F_{n}^3=3F_{n-1}^3+6F_{n-2}^3-3F_{n-3}^3-F_{n-4}^3\)
\(\iff F_{n}^3=-3F_{n+1}^3+6F_{n+2}^3+3F_{n+3}^3-F_{n+4}^3\)
更一般地，用方程式的形式更优雅：
\[F_{n}^3-3F_{n-1}^3-6F_{n-2}^3+3F_{n-3}^3+F_{n-4}^3=0\]
...

说的也是,用方程式的形式来表达线性递推关系更优雅！
\begin{align*}F_{n}-F_{n-1}-F_{n-2}&=0\\
F_{n}^2-2F_{n-1}^2-2F_{n-2}^2+F_{n-3}^2&=0\\
F_{n}^3-3F_{n-1}^3-6F_{n-2}^3+3F_{n-3}^3+F_{n-4}^3&=0\\
F_{n}^4-5F_{n-1}^4-15F_{n-2}^4+15F_{n-3}^4+5F_{n-4}^4-F_{n-5}^4&=0\\
F_{n}^5-8F_{n-1}^5-40F_{n-2}^5+60F_{n-3}^5+40F_{n-4}^5-8F_{n-5}^5-F_{n-6}^5&=0\end{align*}
\((1,−1,−1)\)
\((1,−2,−2,1)\)
\((1,−3,−6,3,1)\)
\((1,−5,−15,15,5,-1)\)
\((1,−8,−40,60,40,-8,-1)\)
找规律：
\begin{align*}
&F_{n}\color{red}{-}F_{n-1}\color{red}{-}F_{n-2}=0\\
&F_{n}^2\color{red}{-2}F_{n-1}^2\color{red}{-2}F_{n-2}^2\color{orange}{+}F_{n-3}^2=0\\
&F_{n}^3\color{red}{-3}F_{n-1}^3\color{red}{-6}F_{n-2}^3\color{orange}{+3}F_{n-3}^3\color{orange}{+}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}{+5}F_{n-4}^4\color{green}{-}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}
不知，大家有没有发现规律呀！
待会再公布！

mathe

2#我忘了每次特征值要变号
对应成多项式就是
$f_2(x)=x^2-x-1,f_3(x)=x^3-2x^2-2x+1,...$
于是${f_{n+2}(x)}/{f_n(-x)}=x^2-L_nx+(-1)^n$

葡萄糖

\begin{align*}
&F_{n}\color{red}{-}F_{n-1}\color{red}{-}F_{n-2}=0\\
&F_{n}^2\color{red}{-2}F_{n-1}^2\color{red}{-2}F_{n-2}^2\color{orange}{+}F_{n-3}^2=0\\
&F_{n}^3\color{red}{-3}F_{n-1}^3\color{red}{-6}F_{n-2}^3\color{orange}{+3}F_{n-3}^3\color{orange}{+}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}{+5}F_{n-4}^4\color{green}{-}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 \\
&F_{n}^6\color{red}{-13}F_{n-1}^6\color{red}{-104}F_{n-2}^6\color{orange}{+260}F_{n-3}^6\color{orange}{+260}F_{n-4}^6\color{green}{-104}F_{n-5}^6\color{green}{-13}F_{n-6}^6\color{Purple}{+1}F_{n-7}^6=0 \\
&F_{n}^7\color{red}{-21}F_{n-1}^7\color{red}{-273}F_{n-2}^7\color{orange}{+1092}F_{n-3}^7\color{orange}{+1820}F_{n-4}^7\color{green}{-1092}F_{n-5}^7\color{green}{-273}F_{n-6}^7\color{purple}{+21}F_{n-7}^7\color{purple}{+1}F_{n-6}^7=0 \\
\end{align*}

\begin{array}{|r|c|l|}
\hline n&1&\color{red}{-F_{n+1}}
&\color{red}{-\frac{F_{n}F_{n+1}}{F_{1}F_{2}}}
&\color{orange}{\frac{F_{n-1}F_{n}F_{n+1}}{F_{1}F_{2}F_{3}}}
&\color{orange}{\begin{pmatrix}
n-2\\
4
\end{pmatrix}_F}
&\color{green}{-\begin{pmatrix}
n-3\\
5
\end{pmatrix}_F}
&\color{green}{-\begin{pmatrix}
n-4\\
6
\end{pmatrix}_F}
&\color{purple}{\begin{pmatrix}
n-5\\
7
\end{pmatrix}_F}
&\color{purple}{\begin{pmatrix}
n-6\\
8
\end{pmatrix}_F}\\
\hline {1}&{1}&\color{red}{-1}&\color{red}{-1}\\  
\hline {2}&{1}&\color{red}{-2}&\color{red}{-2}&\color{orange}{+1}\\  
\hline {3}&{1}&\color{red}{-3}&\color{red}{-6}&\color{orange}{+3}&\color{orange}{+1}\\  
\hline {4}&{1}&\color{red}{-5}&\color{red}{-15}&\color{orange}{+15}&\color{orange}{+5}&\color{green}{-1}\\  
\hline {5}&{1}&\color{red}{-8}&\color{red}{-40}&\color{orange}{+60}&\color{orange}{+40}&\color{green}{-8}&\color{green}{-1}\\
\hline {6}&{1}&\color{red}{-13}&\color{red}{-104}&\color{orange}{+260}&\color{orange}{+260}&\color{green}{-104}&\color{green}{-13}&\color{purple}{+1}\\  
\hline {7}&{1}&\color{red}{-21}&\color{red}{-273}&\color{orange}{+1092}&\color{orange}{+1820}&\color{green}{-1092}&\color{green}{-273}&\color{purple}{+21}&\color{purple}{+1}\\  
\hline \end{array}

\(\color{red}{\begin{pmatrix}
n+1\\
1
\end{pmatrix}_F}=\color{red}{F_{n+1}}\)
\(\color{red}{\begin{pmatrix}
n\\
2
\end{pmatrix}_F}=\color{red}{\frac{F_{n}F_{n+1}}{F_{1}F_{2}}}\)
\(\color{orange}{\begin{pmatrix}
n-1\\
3
\end{pmatrix}_F}=\color{orange}{\frac{F_{n-1}F_{n}F_{n+1}}{F_{1}F_{2}F_{3}}}\)
\(\color{orange}{\begin{pmatrix}
n-2\\
4
\end{pmatrix}_F}=\color{orange}{\frac{F_{n-2}F_{n-1}F_{n}F_{n+1}}{F_{1}F_{2}F_{3}F_{4}}}\)
\(\color{green}{\begin{pmatrix}
n-3\\
5
\end{pmatrix}_F}=\color{green}{\frac{F_{n-3}F_{n-2}F_{n-1}F_{n}F_{n+1}}{F_{1}F_{2}F_{3}F_{4}F_{5}}}\)
\(\color{green}{\begin{pmatrix}
n-4\\
6
\end{pmatrix}_F}=\color{green}{\frac{F_{n-4}F_{n-3}F_{n-2}F_{n-1}F_{n}F_{n+1}}{F_{1}F_{2}F_{3}F_{4}F_{5}F_{6}}}\)
\(\color{purple}{\begin{pmatrix}
n-5\\
7
\end{pmatrix}_F}=\color{purple}{\frac{F_{n-5}F_{n-4}F_{n-3}F_{n-2}F_{n-1}F_{n}F_{n+1}}{F_{1}F_{2}F_{3}F_{4}F_{5}F_{6}F_{7}}}\)
\(\color{purple}{\begin{pmatrix}
n-6\\
8
\end{pmatrix}_F}=\color{purple}{\frac{F_{n-6}F_{n-5}F_{n-4}F_{n-3}F_{n-2}F_{n-1}F_{n}F_{n-1}}{F_{1}F_{2}F_{3}F_{4}F_{5}F_{6}F_{7}F_{8}}}\)

Fibonomial(斐波纳契项)
http://www.maths.surrey.ac.uk/ho ... ci/Fibonomials.html
Fibonomial coefficient(斐波纳契系数)
https://en.wikipedia.org/wiki/Fibonomial_coefficient