卢卡斯数列是斐波那契数和卢卡斯数的推广,以法国数学家爱德华·卢卡斯命名。
给定两个整数P和Q,满足:
![{\displaystyle P^{2}-4Q\neq 0}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/c915cd7e00dad85d89d7817dc8756b0231e1350c)
则第一类卢卡斯数列Un(P,Q)和第二类卢卡斯数列Vn(P,Q)由以下递推关系定义:
![{\displaystyle U_{0}(P,Q)=0\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/ab65e7edf5489f871f8cdbdeeae5992026f52755)
![{\displaystyle U_{1}(P,Q)=1\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/2767d59b5fc386e0da8a6a35eba4779363839f1d)
![{\displaystyle U_{n}(P,Q)=P\cdot U_{n-1}(P,Q)-Q\cdot U_{n-2}(P,Q)\,\,,\,n>1\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/f913a076f4a9361529cc1b13aa34e4003544998f)
以及
![{\displaystyle V_{0}(P,Q)=2\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/002dbc6c1d4985f9ec9c72f24e573aa41f64f30b)
![{\displaystyle V_{1}(P,Q)=P\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/d63802ea912ba7fae75e93a7925e8ec8c3e66584)
![{\displaystyle V_{n}(P,Q)=P\cdot V_{n-1}(P,Q)-Q\cdot V_{n-2}(P,Q)\,\,,\,n>1\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/f1c025265570e4dea3182d84fb139992efa912c9)
卢卡斯数列的特征方程是:
![{\displaystyle x^{2}-Px+Q=0\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/ee4c9d05c963f53aa52587cfbcc0a45fcc1c9650)
它的判别式是
,它的根是:
![{\displaystyle a={\frac {P+{\sqrt {D}}}{2}}\quad ,\quad b={\frac {P-{\sqrt {D}}}{2}}.\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/c0b0ca3b607f915dbe636e053ff66e845b4b3ad7)
注意a和b是不同的,因为
卢卡斯数列的项可以用a和b的项定义如下:
![{\displaystyle U_{n}(P,Q)={\frac {a^{n}-b^{n}}{a-b}}={\frac {a^{n}-b^{n}}{\sqrt {D}}}}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/a0893bbc68f597830daf04c0be23786d51fbea66)
![{\displaystyle V_{n}(P,Q)=a^{n}+b^{n}\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/c811f99862261b2e6d5e257967a6bc91ac70ba39)
从中我们可以推出以下关系:
![{\displaystyle a^{n}={\frac {V_{n}+U_{n}{\sqrt {D}}}{2}}}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/e54d235001ff706176add341e193f3d753c1c6f3)
![{\displaystyle b^{n}={\frac {V_{n}-U_{n}{\sqrt {D}}}{2}}}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/5d2ed69e4fe0d92baadca78ec5913edaf56fe127)
不少斐波那契数和卢卡斯数所满足的关系,在卢卡斯数列中也有类似的形式。例如:
一般 |
P=1, Q=-1
|
![{\displaystyle U_{n}={\frac {V_{n+1}-QV_{n-1}}{P^{2}-4Q}}}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/3ab9b0cc2587e93a1844e2b834ec377679e07a2b) |
|
![{\displaystyle V_{n}=U_{n+1}-QU_{n-1}}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/4eee08284dcecfa78a70c266cc036e280f6c8fe5) |
|
![{\displaystyle U_{2n}=U_{n}V_{n}}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/cc060eff26eab63082b935a4f59c6784d738755b) |
|
![{\displaystyle V_{2n}=V_{n}^{2}-2Q^{n}}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/cf95e7fab570998290688d32240dfe2517721e17) |
|
![{\displaystyle U_{n+m}=U_{n}U_{m+1}-QU_{m}U_{n-1}}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/2c21ffed78d293bfb424a732e608efc9258155e1) |
|
![{\displaystyle V_{n+m}=V_{n}V_{m}-Q^{m}V_{n-m}\,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/2ed1371042d2c9a3eae3c6fa7fdadabd728a3aee) |
|
对于某些P和Q的值,卢卡斯数列有特殊名称:
- Un(1,−1):斐波那契数
- Vn(1,−1):卢卡斯数
- Un(2,−1):佩尔数
- Un(1,−2):Jacobsthal数