1
vote
1answer
71 views

I don't know how to solve equations used in the golden ratio

Today i was reading something from golden ratio and i don't understand how some equations where solved for example: Im told that $\phi_{n+1}=B_{n+1} + \frac {A_n}{B_n}$. What I don't understand is ...
13
votes
4answers
379 views

Converting recursive equations into matrices

How do we convert recursive equations into matrix forms? For instance, consider this recursive equation(Fibonacci Series): $$F_n = F_{n-1} + F_{n-2}$$ And it comes out to be that the following that ...
1
vote
4answers
135 views

Fibonacci Calculation using a larger matrix

So the formula to generate the fibonacci sequence in matrix form is: $$ \begin{pmatrix} 1 & 1 \\ 1 & 0 \\ \end{pmatrix}^n = \begin{pmatrix} F_{n+1} & F_n \\ F_n & ...
3
votes
2answers
148 views

Check if the number is a result of multiplying two fibonacci numbers

Is there a way to check if a given number is a result of multiplication of two fibonacci numbers? Any theory/characteristics allowing to quickly decline some of these given numbers?
1
vote
1answer
255 views

Pisano periods of fibonacci mod

The wikipedia article on Pisano periods utilises the Binet's formula and quadratic residues to find $f(n)$ such that $F_n=f(n) \pmod{p}$ where $p$ is a prime number and $F_n$ is a Fibonacci number. ...
2
votes
2answers
484 views

Using linear algebra, how is the Binet formula (for finding the nth Fibonacci number) derived?

If possible, please refrain from any type of proof besides linear algebra. So, using the recursion formula $F_{n+1} = F_{n-1} + F_n$, for $n\gt 1$, and where $F_0 = 0$ and $F_1 = 1$, and the Fibonacci ...
7
votes
3answers
510 views

Proof of this result related to Fibonacci numbers?

$$\begin{pmatrix}1&1\\1&0\end{pmatrix}^n=\begin{pmatrix}F_{n+1}&F_n\\F_n&F_{n-1}\end{pmatrix}$$ Somebody has any idea how to go about proving this result? I know a proof by ...
21
votes
7answers
923 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 ...