Difference between revisions of "Lucas numbers"
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(Created page with "The Lucas numbers, $L \colon \mathbb{Z} \rightarrow \mathbb{Z}$, is the solution to the following initial value problem: $$L(n+2)=L(n)+L(n+1), \quad L(0)=2, L(1)=1.$$ =See al...") |
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The Lucas numbers, $L \colon \mathbb{Z} \rightarrow \mathbb{Z}$, is the solution to the following initial value problem: | The Lucas numbers, $L \colon \mathbb{Z} \rightarrow \mathbb{Z}$, is the solution to the following initial value problem: | ||
$$L(n+2)=L(n)+L(n+1), \quad L(0)=2, L(1)=1.$$ | $$L(n+2)=L(n)+L(n+1), \quad L(0)=2, L(1)=1.$$ | ||
| + | |||
| + | =Properties= | ||
| + | [[Sum of Lucas numbers]]<br /> | ||
| + | [[Sum of Lucas numbers]]<br /> | ||
| + | [[L(n+1)L(n-1)-L(n)^2=5(-1)^(n+1)]]<br /> | ||
| + | [[L(-n)=(-1)^nL(n)]]<br /> | ||
| + | |||
| + | ==Relationship to Fibonacci numbers== | ||
| + | [[L(n)=F(n+1)+F(n-1)]]<br /> | ||
| + | [[L(n)^2-5F(n)^2=4(-1)^n]]<br /> | ||
| + | [[F(2n)=F(n)L(n)]]<br /> | ||
| + | [[L(n)=F(n+1)+F(n-1)]]<br /> | ||
=See also= | =See also= | ||
Latest revision as of 00:57, 25 May 2017
The Lucas numbers, $L \colon \mathbb{Z} \rightarrow \mathbb{Z}$, is the solution to the following initial value problem: $$L(n+2)=L(n)+L(n+1), \quad L(0)=2, L(1)=1.$$
Properties
Sum of Lucas numbers
Sum of Lucas numbers
L(n+1)L(n-1)-L(n)^2=5(-1)^(n+1)
L(-n)=(-1)^nL(n)
Relationship to Fibonacci numbers
L(n)=F(n+1)+F(n-1)
L(n)^2-5F(n)^2=4(-1)^n
F(2n)=F(n)L(n)
L(n)=F(n+1)+F(n-1)
See also
References
- S.L. Basin and V.E. Hoggatt, Jr.: A Primer on the Fibonacci Sequence Part I (1963)... (previous)... (next) (specifies the following equivalent initial conditions instead: $L(1)=1$ and $L(2)=3$)
