# 卢卡斯数列

## 递推关系

${\displaystyle P^{2}-4Q\neq 0}$

${\displaystyle U_{0}(P,Q)=0\,}$
${\displaystyle U_{1}(P,Q)=1\,}$
${\displaystyle U_{n}(P,Q)=P\cdot U_{n-1}(P,Q)-Q\cdot U_{n-2}(P,Q)\,\,,\,n>1\,}$

${\displaystyle V_{0}(P,Q)=2\,}$
${\displaystyle V_{1}(P,Q)=P\,}$
${\displaystyle V_{n}(P,Q)=P\cdot V_{n-1}(P,Q)-Q\cdot V_{n-2}(P,Q)\,\,,\,n>1\,}$

## 代数关系

${\displaystyle x^{2}-Px+Q=0\,}$

${\displaystyle a={\frac {P+{\sqrt {D}}}{2}}\quad ,\quad b={\frac {P-{\sqrt {D}}}{2}}.\,}$

${\displaystyle U_{n}(P,Q)={\frac {a^{n}-b^{n}}{a-b}}={\frac {a^{n}-b^{n}}{\sqrt {D}}}}$
${\displaystyle V_{n}(P,Q)=a^{n}+b^{n}\,}$

${\displaystyle a^{n}={\frac {V_{n}+U_{n}{\sqrt {D}}}{2}}}$
${\displaystyle b^{n}={\frac {V_{n}-U_{n}{\sqrt {D}}}{2}}}$

## 其他关系

${\displaystyle U_{n}={\frac {V_{n+1}-QV_{n-1}}{P^{2}-4Q}}}$  ${\displaystyle U_{n}={\frac {V_{n+1}+V_{n-1}}{5}}}$
${\displaystyle V_{n}=U_{n+1}-QU_{n-1}}$  ${\displaystyle V_{n}=U_{n+1}+U_{n-1}}$
${\displaystyle U_{2n}=U_{n}V_{n}}$  ${\displaystyle U_{2n}=U_{n}V_{n}}$
${\displaystyle V_{2n}=V_{n}^{2}-2Q^{n}}$  ${\displaystyle V_{2n}=V_{n}^{2}-2(-1)^{n}}$
${\displaystyle U_{n+m}=U_{n}U_{m+1}-QU_{m}U_{n-1}}$  ${\displaystyle U_{n+m}=U_{n}U_{m+1}+U_{m}U_{n-1}}$
${\displaystyle V_{n+m}=V_{n}V_{m}-Q^{m}V_{n-m}\,}$  ${\displaystyle V_{n+m}=V_{n}V_{m}-(-1)^{m}V_{n-m}\,}$

## 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