Difference between revisions of "Fibonacci numbers"

From specialfunctionswiki
Jump to: navigation, search
(Properties)
 
(28 intermediate revisions by the same user not shown)
Line 1: Line 1:
The Fibonacci sequence is defined by
+
__NOTOC__
$$F_{n+2}=F_n+F_{n+1},F_1=F_2=1.$$
+
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=
 
=Properties=
<div class="toccolours mw-collapsible mw-collapsed">
+
[[Limit of quotient of consecutive Fibonacci numbers]]<br />
<strong>Theorem:</strong> The following series holds and converges for all $|x| \leq \dfrac{1}{\varphi}$, where $\varphi$ denotes the [[golden ratio]]:
+
[[Binet's formula]]<br />
$$\dfrac{x}{1-x-x^2} = \displaystyle\sum_{k=1}^{\infty} F_k x^k.$$
+
[[Sum of Fibonacci numbers]]<br />
<div class="mw-collapsible-content">
+
[[Sum of odd indexed Fibonacci numbers]]<br />
<strong>Proof:</strong> proof goes here █
+
[[Sum of even indexed Fibonacci numbers]]<br />
</div>
+
[[Sum of squares of Fibonacci numbers]]<br />
</div>
+
[[Catalan's identity for the Fibonacci sequence]]<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 />
 +
[[F(n)F(n+2)-F(n+1)^2=(-1)^(n+1)]]<br />
 +
[[F(m+h)F(m+k)-F(m)F(m+h+k)=(-1)^mF(h)F(k)]]<br />
 +
[[F(m)=F(k+1)F(m-k)+F(k)F(m-k-1)]]<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 />
  
 
=Videos=
 
=Videos=
[https://www.youtube.com/watch?v=4oyyXC5IzEE The Golden Ratio & Fibonacci Numbers: Fact versus Fiction]<br />
+
[https://www.youtube.com/watch?v=ahXIMUkSXX0 Doodling in Math: Spirals, Fibonacci, and Being a Plant (1 of 3) (21 December 2011)]<br />
[https://www.youtube.com/watch?v=ahXIMUkSXX0 Doodling in Math: Spirals, Fibonacci, and Being a Plant (1 of 3)]<Br />
+
[https://www.youtube.com/watch?v=4oyyXC5IzEE The Golden Ratio & Fibonacci Numbers: Fact versus Fiction (11 December 2012)]<br />
[https://www.youtube.com/watch?v=Nu-lW-Ifyec Fibonacci mystery]<br />
+
[https://www.youtube.com/watch?v=Nu-lW-Ifyec Fibonacci mystery (18 September 2013)]<br />
  
 
=See also=
 
=See also=
 +
[[Fibonacci zeta function]]<br />
 
[[Golden ratio]]<br />
 
[[Golden ratio]]<br />
 
[[Reciprocal Fibonacci constant]]<br />
 
[[Reciprocal Fibonacci constant]]<br />
Line 23: Line 39:
 
=External links=
 
=External links=
 
[http://www.fq.math.ca/ The Fibonacci Quarterly]<br />
 
[http://www.fq.math.ca/ The Fibonacci Quarterly]<br />
[http://matheducators.stackexchange.com/questions/2021/what-interesting-properties-of-the-fibonacci-sequence-can-i-share-when-introduci]<br />
+
[http://matheducators.stackexchange.com/questions/2021/what-interesting-properties-of-the-fibonacci-sequence-can-i-share-when-introduci "What interesting properties of the Fibonacci sequence can I share when introducing sequences?"]<br />
 +
 
 +
=References=
 +
* {{PaperReference|Sur la série des inverse de nombres de Fibonacci|1899|Edmund Landau|next=Limit of quotient of consecutive Fibonacci numbers}}
 +
* {{PaperReference|A Primer on the Fibonacci Sequence Part I|1963|S.L. Basin|author2=V.E. Hoggatt, Jr.|next=Lucas numbers}}
 +
* {{PaperReference|On Identities for Fibonacci Numbers|1963|David Zeitlin}}
 +
* {{PaperReference|On a General Fibonacci Identity|1965|John H. Halton|next=Binet's formula}}
 +
* {{PaperReference|The Fibonacci Zeta Function|1976|Maruti Ram Murty|next=Fibonacci zeta function}}
 +
 
 +
[[Category:SpecialFunction]]

Latest revision as of 13:12, 1 August 2018

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)
F(n)F(n+2)-F(n+1)^2=(-1)^(n+1)
F(m+h)F(m+k)-F(m)F(m+h+k)=(-1)^mF(h)F(k)
F(m)=F(k+1)F(m-k)+F(k)F(m-k-1)

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?"

References