Fibonacci numbers

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)