Lucas numbers

From specialfunctionswiki
Revision as of 00:57, 25 May 2017 by Tom (talk | contribs)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

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