Difference between revisions of "Fibonacci numbers"
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| − | The Fibonacci | + | 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= | ||
[[Limit of quotient of consecutive Fibonacci numbers]]<br /> | [[Limit of quotient of consecutive Fibonacci numbers]]<br /> | ||
[[Binet's formula]]<br /> | [[Binet's formula]]<br /> | ||
| + | [[Sum of Fibonacci numbers]]<br /> | ||
| + | [[Sum of odd indexed Fibonacci numbers]]<br /> | ||
| + | [[Sum of even indexed Fibonacci numbers]]<br /> | ||
| + | [[Sum of squares of Fibonacci numbers]]<br /> | ||
| + | [[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= | + | [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= | + | [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 /> | ||
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=References= | =References= | ||
* {{PaperReference|Sur la série des inverse de nombres de Fibonacci|1899|Edmund Landau|next=Limit of quotient of consecutive Fibonacci numbers}} | * {{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]] | [[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
- Edmund Landau: Sur la série des inverse de nombres de Fibonacci (1899)... (next)
- S.L. Basin and V.E. Hoggatt, Jr.: A Primer on the Fibonacci Sequence Part I (1963)... (next)
- David Zeitlin: On Identities for Fibonacci Numbers (1963)
- John H. Halton: On a General Fibonacci Identity (1965)... (next)
- Maruti Ram Murty: The Fibonacci Zeta Function (1976)... (next)
