Fibonacci Number

Let's see some different topic today . Its time to have a fresh look at some of the beautiful series .

Fibonacci Number : http://en.wikipedia.org/wiki/Fibonacci_number Here you can have ample info but I want to give some candy information in a group :-

1. (F(n)) | F(n) = F(n-1)+F(n-2) , F(0) = 0 , F(1) = 1 . Its the definition of this sequence called Fibonacci 

     sequence. 

2. Its an infinite diverging sequence :-

    Lt (n → ∞ , F(n+1)/F(n)) = 1+1/(1+1/( ... )) = y (say) = 1/1+y => y^2+y-1 = 0 => y =  φ >1 thus it is 

    divergent.   

3. gcd(F(n),F(m)) = F(gcd(n,m)) 

4. F(n) = Σ( k = 0 to floor((n-1)/2) , C(n-k-1,k)) the second diagonal sum of Pascal triangle.   

5. (F(n+1) ; F(n)) = (1 1 ; 1 0)^n (1 ; 0)

6. F(m)F(n) + F(m-1)F(n-1) = F(m+n-1) [ From above relation ]

7. G(x) =  generating function = x / (1 - x - x^2) [ From relation 1]

8. F(0) + F(1) + ... + F(n) = F(n+2) - 1

You can go further by seeing this http://en.wikipedia.org/wiki/Pisano_period