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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__NOTOC__
 
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:
 
$$L(n+2)=L(n)+L(n+1), \quad L(0)=2, L(1)=1.$$
 
$$L(n+2)=L(n)+L(n+1), \quad L(0)=2, L(1)=1.$$
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=Properties=
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[[Sum of Lucas numbers]]<br />
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[[Sum of Lucas numbers]]<br />
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[[L(n+1)L(n-1)-L(n)^2=5(-1)^(n+1)]]<br />
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[[L(-n)=(-1)^nL(n)]]<br />
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==Relationship to Fibonacci numbers==
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[[L(n)=F(n+1)+F(n-1)]]<br />
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[[L(n)^2-5F(n)^2=4(-1)^n]]<br />
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[[F(2n)=F(n)L(n)]]<br />
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[[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

Fibonacci numbers

References