Questions tagged [fibonacci-numbers]

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 $F_1=1$.

The $n$th Fibonacci number $F_n$ is defined recursively, by

$$F_n = F_{n - 1} + F_{n - 2}$$

for $n > 1$, and $F_0 = 0,\; F_1 = 1$. There is a closed form expression, namely

$$F_n = \frac{\varphi^n - (1 - \varphi)^n}{\sqrt{5}}$$

where the golden ratio $\varphi$ is equal to $\frac{1 + \sqrt{5}}{2}$.

Combinatorial identities involving the Fibonacci numbers have been extensively studied, and the numbers arise frequently in nature and in popular culture.

Reference: Fibonacci number.

2003 questions
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Proof that the ratio between the logs of the product and LCM of the Fibonacci numbers converges to $\frac{\pi^2}{6}$

I came across this amazing fact on Twitter. $$\lim_{n\to\infty} \frac{\log\left(F_1 \cdots F_n\right)}{\log \text{LCM}\left(F_1, \ldots, F_n\right)} = \frac{\pi^2}{6}$$ where $F_i$ is the $i$th Fibonacci number and LCM = Least Common Multiple. This…
27
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How to prove that the Binet formula gives the terms of the Fibonacci Sequence?

This formula provides the $n$th term in the Fibonacci Sequence, and is defined using the recurrence formula: $u_n = u_{n − 1} + u_{n − 2}$, for $n > 1$, where $u_0 = 0$ and $u_1 = 1$. Show that $$u_n = \frac{(1 + \sqrt{5})^n - (1 -…
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Is the Fibonacci sequence exponential?

I could not find any information on this online so I thought I'd make a question about this. If we take the Fibonacci sequence $F_n = F_{n-1} + F_{n-2}$, is this growing exponentially? Or perhaps if we consider it as a function $F(x) = F(x-1) +…
Stijn
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The Fibonacci sum $\sum_{n=0}^\infty \frac{1}{F_{2^n}}$ generalized

The evaluation, $$\sum_{n=0}^\infty \frac{1}{F_{2^n}}=\frac{7-\sqrt{5}}{2}=\left(\frac{1-\sqrt{5}}{2}\right)^3+\left(\frac{1+\sqrt{5}}{2}\right)^2$$ was recently asked in a post by Chris here. I like generalizations, and it turns out this is not a…
26
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Parabolas in sequences of digits from the Fibonacci sequence

In preperation for an exam, I was studying Haskell. Therefore I was solving an old assignment where you had to define the fibonacci series. After solving the task (see 1] for source code) and reviewing the result, I found a rather interesting…
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Applications of the Fibonacci sequence

The Fibonacci sequence is very well known, and is often explained with a story about how many rabbits there are after $n$ generations if they each produce a new pair every generation. Is there any other reason you would care about the Fibonacci…
Gordon Gustafson
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Combinatorial proof of a Fibonacci identity: $n F_1 + (n-1)F_2 + \cdots + F_n = F_{n+4} - n - 3.$

Does anyone know a combinatorial proof of the following identity, where $F_n$ is the $n$th Fibonacci number? $$n F_1 + (n-1)F_2 + \cdots + F_n = F_{n+4} - n - 3$$ It's not in the place I thought it most likely to appear: Benjamin and Quinn's Proofs…
Mike Spivey
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Fibonacci-related sum

Related to this question Find a solution for $f\left(\frac{1}{x}\right)+f(x+1)=x$, what is this sum: $$\sum_{n=1}^{\infty}(-1)^n\left(\frac{F_n}{F_{n+1}}-\frac1{\phi}\right)$$ where $F_n$ is the $n$th Fibonacci number and $\phi=\frac{1+\sqrt{5}}2$…
Empy2
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Series involving Fibonacci Numbers: $\sum_{k=1}^\infty \frac{1}{F_kF_{k+1}}$

I will start my question with a bit of information that I think may be helpful to potential answerers. If you don't want to read it, skip down to the question. BACKGROUND: I'm investigating series in the form $$\Phi_n(x):=\sum_{k=1}^\infty…
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4 answers

Do Fibonacci numbers form a complete residue system in every modulus?

I want to show that: $$\forall x,m\ \exists n:x\equiv_mF_n$$ I assume that one can prove this by the pigeonhole principle, but I couldn't manage to find a series of $m+1$ numbers that each want to occupy a different number.
LionCoder
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Fibonacci addition law $F_{n+m} = F_{n-1}F_m + F_n F_{m+1}$

Question: Let $F_n$ the sequence of Fibonacci numbers, given by $F_0 = 0, F_1 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n \geq 2$. Show for $n, m \in \mathbb{N}$: $$F_{n+m} = F_{n-1}F_m + F_n F_{m+1}$$ My (very limited) attempt so far: after creating…
ghshtalt
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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, $\frac{1}{3}$ in base 10 is $0.33333...$, in base 5 it's $0.131313...$, and in base 3 it's…
Joe
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Continued fraction involving Fibonacci sequence

What is the limit of the continued fraction: $$\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{5+\cfrac{1}{8+\cdots}}}}}}\ $$ that involves the Fibonacci sequence terms as denominators? I've been looking for this specific continued…
22
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Asymptotic behaviour of sum

I would like to evaluate the number $c$ given by $$ c = \lim_{m\to\infty} \frac{1}{\log m}\sum_{n=1}^m \frac{1}{n^2 \sin^2(\pi n \tau)} $$ where $\tau = (1+\sqrt{5})/2$. My attempt: my guess was this sum would be dominated by the terms for which…
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Proving there is no non-abelian finite simple group of order a Fibonacci number

"Prove there does not exist a finite simple non-abelian group of order of a Fibonacci number" I would like to answer the above question, but I currently have few ideas of where to begin. I understand we will likely only be using results about the…