Difference between revisions of "Fibonacci numbers"
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[[Catalan's identity for the Fibonacci sequence]]<br /> | [[Catalan's identity for the Fibonacci sequence]]<br /> | ||
[[F(n+1)F(n-1)-F(n)^2=(-1)^n]]<br /> | [[F(n+1)F(n-1)-F(n)^2=(-1)^n]]<br /> | ||
| + | [[F(2n+1)=F(n+1)^2+F(n)^2]]<br /> | ||
| + | [[F(2n)=F(n+1)^2-F(n-1)^2]]<br /> | ||
| + | [[F(n+m+1)=F(n+1)F(m+1)+F(n)F(m)]]<br /> | ||
| + | [[F(-n)=(-1)^(n+1)F(n)]]<br /> | ||
| + | |||
| + | ==Relationship with Lucas numbers== | ||
| + | [[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 00:51, 25 May 2017
The Fibonacci numbers, $F \colon \mathbb{Z} \rightarrow \mathbb{Z}$, is the solution of the following initial value problem: $$F(n+2)=F(n)+F(n+1), \quad F(0)=0, F(1)=1.$$ Often, $F(n)$ is written with a subscript: $F_n$.
Properties
Limit of quotient of consecutive Fibonacci numbers
Binet's formula
Sum of Fibonacci numbers
Sum of odd indexed Fibonacci numbers
Sum of even indexed Fibonacci numbers
Sum of squares of Fibonacci numbers
Catalan's identity for the Fibonacci sequence
F(n+1)F(n-1)-F(n)^2=(-1)^n
F(2n+1)=F(n+1)^2+F(n)^2
F(2n)=F(n+1)^2-F(n-1)^2
F(n+m+1)=F(n+1)F(m+1)+F(n)F(m)
F(-n)=(-1)^(n+1)F(n)
Relationship with Lucas numbers
L(n)^2-5F(n)^2=4(-1)^n
F(2n)=F(n)L(n)
L(n)=F(n+1)+F(n-1)
Videos
Doodling in Math: Spirals, Fibonacci, and Being a Plant (1 of 3) (21 December 2011)
The Golden Ratio & Fibonacci Numbers: Fact versus Fiction (11 December 2012)
Fibonacci mystery (18 September 2013)
See also
Fibonacci zeta function
Golden ratio
Reciprocal Fibonacci constant
Lucas numbers
External links
The Fibonacci Quarterly
"What interesting properties of the Fibonacci sequence can I share when introducing sequences?"
