Tagged Questions

5
votes
3answers
131 views

Summation of Fibonacci numbers.

Let $f_n$ be the sequence of Fibonacci numbers. We need to show that $$\sum_{n\ge0} f_n x^n = \dfrac{1}{1-x-x^2}$$ I remember a solution when we are using the generating functions like: $f(x) = F_0 ...
1
vote
1answer
122 views

Fibonacci Generating Function of a Complex Variable

So I'm doing work on the Fibonacci Numbers, and I came across this problem for the generating function for the recursive fibonacci numbers. I have two questions: 1. Why is it useful to use a ...
3
votes
1answer
152 views

Generating Function of Even Fibonacci

I was posed the following question recently on an exam: Determine the generating function of the even-indexed Fibonacci numbers $F_{2n}$ given that the generating function of Fibonacci numbers is ...
3
votes
1answer
195 views

Expanding the generating function of the Fibonacci numbers to find a cute formula

$F_0=1$, $F_1=1$, $F_n=F_{n-1}+F_{n-2}$. The generating function is $-\frac{1}{x^2+x-1}$. I have to expand it to prove that $F_n=\sum_k\binom{k}{n-k}$. Could you help me please?
3
votes
1answer
366 views

On the generating function of the Fibonacci numbers

Let's define the Fibonacci numbers as $F_0=1$, $F_1=1$ and $F_n=F_{n-1}+F_{n-2}$. Using this recurrence I was able to calculate the generating function of the Fibonacci numbers to be ...
5
votes
0answers
233 views

Construction of generating function from identity

I am trying to solve identity involving binomials and fibbonaci numbers by using generating functions: $$\sum_{k=0}^n{n \choose k}{n+k\choose k}f_{k+1}=\sum_{k=0}^n{n \choose k}{n+k\choose ...
20
votes
7answers
656 views

How are we able to calculate specific numbers in the Fibonacci Sequence?

I was reading up on the Fibonacci Sequence when I've noticed some were able to calculate specific numbers. So far I've only figured out creating an array and counting to the value, which is incredibly ...