定义:Fibonacci数列:f1=1,f2=1f_1=1,f_2=1f1=1,f2=1,且对n≥3n\ge3n≥3,fn=fn−1+fn−2f_n=f_{n-1}+f_{n-2}fn=fn−1+fn−2
约定:f0=0f_0=0f0=0.
推论:Fibonacci数fn=(1+52)n−(1−52)n5f_n=\dfrac{({\dfrac{1+\sqrt 5}{2})^{n}}-{(\dfrac{1-\sqrt 5}{2})^{n}}}{\sqrt 5}fn=5(21+5)n−(21−5)n
定理:Fibonacci数列的一些性质:f2n=fn2+2fn−1fnf_{2n}=f_{n}^{2}+2f_{n-1}f_nf2n=fn2+2fn−1fn fn−2+fn+2=3fnf_{n-2}+f_{n+2}=3f_{n}fn−2+fn+2=3fn fn+1fn−1−fn2=(−1)nf_{n+1}f_{n-1}-f_{n}^{2}=(-1)^{n}fn+1fn−1−fn2=(−1)n fm+n=fmfn+1+fnfm−1f_{m+n}=f_{m}f_{n+1}+f_{n}f_{m-1}fm+n=fmfn+1+fnfm−1 ∑k=1nfk=fn+2−1\sum_{k=1}^{n}{f_{k}}=f_{n+2}-1k=1∑nfk=fn+2−1 ∑k=1nf2k−1=f2n−f2+f1\sum_{k=1}^{n}{f_{2k-1}}=f_{2n}-f_{2}+f_{1}k=1∑nf2k−1=f2n−f2+f1 ∑k=1nf2n=f2n+1−f1\sum_{k=1}^{n}{f_{2n}}=f_{2n+1}-f_{1}k=1∑nf2n=f2n+1−f1 ∑k=1nfk2=fnfn+1\sum_{k=1}^{n}{f_{k}^{2}}=f_{n}f_{n+1}k=1∑nfk2=fnfn+1 ∑k=12n−1fkfk+1=f2n2\sum_{k=1}^{2n-1}{f_{k}f_{k+1}=f_{2n}^{2}}k=1∑2n−1fkfk+1=f2n2
定义:黄金分割比(Golden ratio)φ:=limn→∞Fn+1Fn=1+52\varphi\coloneqq\lim\limits_{n\rightarrow\infty}{\dfrac{F_{n+1}}{F_n}}=\dfrac{1+\sqrt5}{2}φ:=n→∞limFnFn+1=21+5
定义:Lucas数列:L1=1,L2=3L_1=1,L_2=3L1=1,L2=3,且对n≥3n\ge3n≥3,Ln=Ln−1+Ln−2L_n=L_{n-1}+L_{n-2}Ln=Ln−1+Ln−2
Ln=(1+52)n−(1−52)nL_n=(\dfrac{1+\sqrt 5}{2})^{n}-(\dfrac{1-\sqrt 5}{2})^{n}Ln=(21+5)n−(21−5)n
Lucas数列也具有Fibonacci数列类似的性质.
定理:Zeckendorf表示:对于任意n∈Nn\in\Nn∈N,存在唯一的一组Fibonacci子列,使得n=∑i=0kFcin=\displaystyle\sum_{i=0}^{k}{F_{c_i}}n=i=0∑kFci,其中ci≥2,ci+1>ci+1c_i\ge2,c_{i+1}>c_i+1ci≥2,ci+1>ci+1
定义:广义Fibonacci数列:g1=a,g2=bg_1=a,g_2=bg1=a,g2=b,且对n≥3n\ge3n≥3,gn=gn−1+gn−2g_n=g_{n-1}+g_{n-2}gn=gn−1+gn−2
gn=afn−2+bfn−1n≥3g_n=af_{n-2}+bf_{n-1}\quad n\ge3gn=afn−2+bfn−1n≥3
当nnn为负整数时,递归定义Fibonacci数fn−2:=fn−fn−1f_{n-2}\coloneqq f_{n}-f_{n-1}fn−2:=fn−fn−1
f−n=(−1)n+1fnf_{-n}={(-1)}^{n+1}f_nf−n=(−1)n+1fn