# Tagged Questions

Questions on the Fibonacci numbers, a special sequence of integers that satisfy the recurrence $F_n=F_{n-1}+F_{n-2}$ with the initial conditions $F_0=0$ and $F_1=1$.

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### Rate of Convergence vs Radius of Convergence

What is the difference between finding the 'rate of convergence' and the radius of convergence'? The question I am trying to solve here is to find the rate of convergence of the ratio of Fibonacci ...
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### On $4n+1 = x^2, 5n+1 = y^2$ and the Fibonacci numbers

While answering this question, I decided to look at the particular case, $$4n+1 = x^2\\5n+1 = y^2$$ to be solved simultaneously. The solutions for $n,x,y$ are A157459, A007805, and A049629, ...
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### Does 1/4 eventually repeat in Fibonacci coding?

In a regular, positional number system (like our decimal numbers), every rational number ends in a repeating sequence (even if it's just 0). For example, 1/3 in base 10 is 0.33333..., in base 5 it's ...
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### Do all rational numbers repeat in Fibonacci coding?

In a regular, positional number system (like our decimal numbers), every rational number ends in a repeating sequence (even if it's just 0). For example, 1/3 in base 10 is 0.33333..., in base 5 it's ...
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### Need help calculating probability…

First time here, so I hope you'll not get too frustrated if I make any etiquette mistakes for this forum. So here's my question. I know there are snow day calculators out there, but I'm trying to ...
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### Is it rational or not?

I have two interesting question : Is this number rational or not: $$0.F_{1}F_{2}F_{3}...$$, where $F_{i}$ - Fibonacci number. And is this number rational or not: $$0.p_{1}p_{2}...$$
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### Find $\mathrm{lcm}(F_{m},F_{n},F_{k})$.

So consider three Fibonacci numbers. My question is: Find $\mathrm{lcm}(F_{m},F_{n},F_{k})$, where lcm is least common multiple.
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### Is the “Fibonacci square tiling” of Fibonacci-sided rectangles always optimal?

Is an optimal square tiling of a rectangle with side lengths of successive Fibonacci numbers always the sequence of Fibonacci numbers, as in the picture below?
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### Convergence of Fibonacci quotients

Let $F_n=F_{n-1}+F_{n-2},~ F_0=0,~F_1=1~$ be the Fibonacci numbers. Then it is well known that $\lim_n \frac{F_{n+1}}{F_n}=\frac{1+\sqrt{5}}{2}$. However, many textbooks proved the above by using ...
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### Fibonacci identity $F_{n+1}^2 - (F_{n+1}F_n) - F_n^2 = (-1)^n$

I am trying to prove Let $F_n$ be the $n$th Fibonacci number. Then $F_{n+1}^2 - F_{n+1}F_n - F_n^2 = (-1)^n$ I am not sure where to start with this.
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### Limit as $n \to \infty$ of $\frac{x_{n+1}}{x_n}$ in the Fibonacci sequence

Given that $x_n = \frac{\sqrt{5}+5}{10} (\frac{1 + \sqrt{5}}{2})^n + \frac{5 - \sqrt{5}}{10}(\frac{1 - \sqrt{5}}{2})^n$ How do you show that $\lim_{n \to \infty} \frac{x_{n+1}}{x_n}$ is the ...
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### Proof concerning Fibonacci and recursively defined sequences

The series $(a_n)_{n\in\mathbb{N}}$ is given through $$a_1=1,\quad a_2=\frac{1}{2},\quad a_{n+2}=a_na_{n+1} \quad\text{ for } n\geq1.$$ I want to show that $$a_n = 2^{-f_{n-1}}$$ whereas $\,f_n$ ...
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### Seeking a combinatorial proof $F_{mn}$ always a multiple of $F_m$

I would appreciate if somebody could help me with the following problem Q: Let $F_n$ the sequence of Fibonacci numbers, given by $F_1 = 1, F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n \geq 2$ ...
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### Integral representation for Fibonacci's numbers

We know that, for example, the Gamma function is a perfect integral representation for the factorial $n!$ for a natural number $n$. $$\Gamma[n] = \int_0^{+\infty} t^{n-1}e^{-t}\text{d}t = (n-1)!$$ ...
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### Is there a proven way to calculate the entry point(first occurence) of a factor m, in the Fibonacci sequence?

I saw a comment at the OEIS website for the sequence of entry points, of Fibonacci factors. https://oeis.org/A001177 It referenced a paper by Mark Renault in 1996, with the quote from OEIS: http://...
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### $|x^2-xy-y^2|=1$ implies that $x=\pm F_{n+1},\; y=\pm F_n$

So I've proved that $A= \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix} \implies A^n= \begin{pmatrix} F_{n+1} & F_n\\ F_n & F_{n-1} \end{pmatrix}$ for Fibonacci numbers $F_i$. I'm ...
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### How to apply geometric series concepts into these numbers?

This is a basic level question and it is homework for someone that I am trying to help out. It is indicated as a Fibonacci puzzle. But I am not able to fit the numbers into a general geometric formula....
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### Proving a Fibonacci identity: $F_{2n} = F_n (F_{n+1} + F_{n-1})$

$$F_{2n} = F_n (F_{n+1} + F_{n-1})$$ I'm so stuck. I've used the definition of Fibonacci to change $F_{2n+2}$ into $F_{2n+1} + F_{2n}$. Can't use other properties, only the inductive hypothesis and ...
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### Clarification on tribonacci numbers exercise

From what I know the Tribonacci sequence is given by: T(n) = T(n-1) + T(n-2) + T(n-3) My book says that "We can show by induction that for large enough n, the Fibonacci numbers satisfy the ...
$F_{i}$ denotes the $i^{th}$ Fibonacci number, but what does it mean when there are 2 subscripts, $F_{ij}$? Context: show that $F_{i}$|$F_{ij}$ (where $i$ and $j$ are positive integers) Thanks