Difference between revisions of "Lucas numbers"

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(Properties)
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=Properties=
 
=Properties=
 
[[Sum of Lucas numbers]]<br />
 
[[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 />
 
[[L(n)=F(n+1)+F(n-1)]]<br />
  

Revision as of 00:55, 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

Fibonacci numbers

References