# 卢卡斯数

${\displaystyle L_{n}=L(n)={\begin{cases}2&{\mbox{if }}n=0;\\1&{\mbox{if }}n=1;\\L(n-1)+L(n-2)&{\mbox{if }}n>1.\\\end{cases}}}$

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ... （OEIS數列A000032

## 延伸到负数

(... -11, 7, -4, 3, -1, 2, 1, 3, 4, 7, 11, ...)

• ${\displaystyle L_{-n}=(-1)^{n}L_{n}.\!}$

## 与斐波那契数的关系

• ${\displaystyle \,L_{n}=F_{n-1}+F_{n+1}}$
• ${\displaystyle \,L_{n}^{2}=5F_{n}^{2}+4(-1)^{n}}$，因此，当${\displaystyle n\,}$趋近于无穷大时，${\displaystyle L_{n} \over F_{n}\,}$趋近于${\displaystyle {\sqrt {5}}\,}$
• ${\displaystyle \,F_{2n}=L_{n}F_{n}}$
• ${\displaystyle \,F_{n}={L_{n-1}+L_{n+1} \over 5}}$

${\displaystyle L_{n}=\varphi ^{n}+(1-\varphi )^{n}=\varphi ^{n}+(-\varphi )^{-n}=\left({1+{\sqrt {5}} \over 2}\right)^{n}+\left({1-{\sqrt {5}} \over 2}\right)^{n}\,,}$

## 卢卡斯素数

2, 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, ... （OEIS數列A005479

## 参考文献

• Hoggatt, V. E. Jr. The Fibonacci and Lucas numbers. Boston, MA: Houghton Mifflin, 1969.
• Hrant Arakelian. Mathematics and History of the Golden Section, Logos 2014, 404 p. ISBN 978-5-98704-663-0 (rus.).