Questions tagged [fibonacci-numbers]

Question 1

I've been exploring a question about composite Fibonacci numbers and I'm not sure if my findings are new or if this is a known problem. Definitions Let $F_n$ be the $n$-th Fibonacci number ($F_1=1, ...

Question 2

Consider the sequence $(x_n)_{n\ge 0}$ defined by $$x_0=1,\;\;\;x_1=1,$$ and for $n\ge 1$ $$x_{n+1}=\sum_{0\le i\lt j\le n}x_ix_j\;\;\;\;\;\;(1)$$ So $x_{n+1}$ is the sum of all pairwise products of ...

Question 3

As in the heading, I'm trying to write up a proof that the quotients of all Fibonacci numbers is not dense in $\mathbb{R_+}$. This is what I have come up with and would like to know if it's correct. ...

Question 4

Compute the value of $$\sqrt{1 + F_2\sqrt{1 + F_4\sqrt{1 + F_6\sqrt{1 + F_{2n}\ldots}}}}$$ where $F_n$ denotes the $n$-th Fibonacci number with $F_0 = 0$, $F_1 = 1$. This is a problem from a sheet ...

Question 5

I came across the following integral in an integration bee paper $$I=\int_{-\infty}^\infty \prod_{n=1}^\infty \frac{\sin(\frac{x}{F_{2n}})}{\frac{x}{F_{2n}}}dx$$ where $F_n$ is the $n$th Fibonacci ...

Question 6

Define $f_1=f_2=1$, $f_{n+2}=f_{n+1}+f_n$, and $p$ is an odd prime. Show that $$p\;\mid\;(p-1)!\sum_{i=1}^{p-1}\frac{f_i}{i}$$ I saw this in a sample of high school contest, but I don't know how to ...

Question 7

I noticed that for a Fibonacci sequence starting with seeds $(2,1)$, there is an awful amount of primes in the first $20$ elements of the sequence ($11$ primes), far more than $(0,1)$'s prime density. ...

Question 8

Using the following python program, then we find for large $N$, $$\sum_{1}^{N}{(\phi - \frac{F_{n+1}}{F_n})} \approx 0.3184529640745031$$ But for the life of me I cannot find an exact value for this, ...

Question 9

This reference contains two similar series, which are \begin{align} \arctan \frac{1}{F_{2n + 1}} = \arctan \frac{1}{F_{2n}} &- \arctan \frac{1}{F_{2n + 2}} \\[6pt] \Longrightarrow \qquad \sum_{n = ...

Question 10

I was investigating the product $$\prod_{i = 0}^{\infty}\frac{p_i}{p_i - 1},$$ where $p_i$ is the $i$th prime number (and $p_0 = 2$). After failing to determine whether it diverges on my own, I found ...

Question 11

I have no idea how to prove if the following series converge: $$ \sum_{n=1}^{+\infty}\frac{(n\varphi-\lfloor n\varphi\rfloor)^n}{n} $$ where $\varphi$ is the golden ratio. None criterion clearly ...

Question 12

Fibonacci Algorithms This question came to me when I learned today that the Fibonacci sequence has a really neat $2\times 2$ matrix power defnition which enables a newbie to compute $n$th Fibonacci ...

Question 13

Let $f(x) = 2x^2 + x - 1$ and $$ f^n(x) = (\underbrace{f \circ \dotsb \circ f}_{n \text{ copies of } f})(x). $$ Given $n \ge 1$, what is the number of real roots to $f^n(x) = 0$? Note: This is an ...

Question 14

While I was reading a book, I found the identity \begin{align*} \sum_{k = 0}^{m} (-1)^{[(m - k)/2]}\mathscr{F}_{m}^{k} F_{n + k}^{m - 1} = 0 , \end{align*} where the $ [x] $ denotes the integer that ...

Question 15

Let $\{F_n\}$ be the Fibonacci sequence defined by $$ F_0 = 0, \quad F_1 = 1, \quad F_{n+2} = F_{n+1} + F_n. $$ Initially, the problem I wanted to solve was simply to prove that $$ 5F_{n+2} + F_n $$ ...

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Questions tagged [fibonacci-numbers]

A conjecture on representing composite Fibonacci numbers

A super-exponential sequence related to the Fibonacci sequence

Proving that the set of the quotients of Fibonacci numbers is not dense in the positive reals

How to calculate the iterations of Fibonacci Sequence under square roots

Integrating $\int_{-\infty}^\infty\prod_{n=1}^\infty\frac{\sin(x/F_{2n})}{x/F_{2n}}dx$, where $F_n$ is the $n$th Fibonacci number

Showing that $p$ divides $(p-1)!\sum_{i=1}^{p-1}\frac{f_i}{i}$, where $f_i$ is the $i$-th Fibonacci number, and $p$ is an odd prime

Fibonacci Primes

Finding the value of $\sum_{n=1}^{\infty}{\phi - \frac{F_{n+1}}{F_n}}$ where $\phi = \frac{1+\sqrt5}{2}$, $F_{n+2}=F_{n+1}+F_n$, and $F_1=1, F_2=1$ [duplicate]

Showing $\sum_{n=1}^\infty\arctan\frac1{F_{2n + 1}}=\sum_{n=0}^\infty\arctan\frac1{\sqrt5F_{2n + 1}}$, without showing each sum is $\pi/4$

Do the Fibonacci numbers appear in the products $\prod_{i=0}^N\frac{p_i}{p_i-1}$, with $p_i$ the $i$-th prime, or is it just a coincidence?

Convergence of a series involving Fibonacci numbers

A Fibonacci-like phenomenon seems to be happening amongst the primes up to an additive term of $+1, -1,$ or $\pm 1$ in each sequence value.

The number of real roots of $f^n(x) = 0$ where $f(x) = 2x^2 + x - 1$

$ F_{n} $ is Fibonacci squence, how to prove $ \sum_{k = 0}^{m} (-1)^{k(k + 1)/2}\mathscr{F}_{m}^{k} F_{n + k}^{m - 1} = 0 $?

$v_5(5F_{n+2} + F_n)$ the largest exponent $k$ such that $5^k \mid (5F_{n+2} + F_n)$

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