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Questions tagged [golden-ratio]

Questions relating to the golden ratio $\varphi = \frac{1+\sqrt{5}}{2}$

3
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2answers
130 views

Studying on this sum interesting $\sum_{n=1}^{\infty}\frac{{2n \choose n }}{4^n n}$

I was studying this particular sum $$\sum_{n=1}^{\infty}\frac{{2n \choose n}}{4^n n}$$ and eventually I ended up with is sum $(1)$ $$\sum_{n=1}^{\infty}\frac{{2n \choose n}}{(-4\phi)^n}\cdot\frac{1}{(...
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1answer
89 views

Harmonic number and the golden ratio

I just don't understand how $(1)$ can have this simple closed form. $$\sum_{k=1}^{\infty}\frac{{2k \choose k}}{(-16)^k}[H_k-H_{k+1}]=\phi^{-6}\tag1$$ Where $\phi=\frac{1+\sqrt{5}}{2}$, is the ...
26
votes
1answer
797 views

What is the maximum value of this nested radical?

I was experimenting on Desmos (as usual), in particular infinite recursions and series. Here is one that was of interest: What is the maximum value of $$F_\infty=\sqrt{\frac{x}{x+\sqrt{\frac{x^2}{x-...
9
votes
3answers
754 views

A Series For the Golden Ratio

Question: Can we show that $$\phi=\frac{1}{2}+\frac{11}{2}\sum_{n=0}^\infty\frac{(2n)!}{5^{3n+1}(n!)^2} $$; where $\phi={1+\sqrt{5} \above 1.5pt 2}$ is the golden ratio ? Some background and ...
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2answers
51 views

Why is $F_n = r^n$ a solution of the difference equation if $r$ satisfies $r^2-r-1=0$?

The following is from p.4 of https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/ch3.pdf The terms in the Fibonacci sequence are uniquely determined by the linear difference equation $$...
2
votes
3answers
142 views

How to show that $\sum_{n=1}^{\infty}\frac{\phi^{2n}}{n^2{2n \choose n}}=\frac{9}{50}\pi^2$

Given:$$\sum_{n=1}^{\infty}\frac{\phi^{2n}}{n^2{2n \choose n}}=\frac{9}{50}\pi^2$$ Where $\phi=\frac{\sqrt{5}+1}{2}$ How can I we show that the above sum is correct? I have checked numerically, it ...
4
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0answers
46 views

Functional Equation involving Golden Ratio [duplicate]

Find all $h:\mathbb{N}\rightarrow \mathbb{N}$ such that $$h(h(n)) + h(n+1) = n+2$$ I tried this, but wasn't able to make any progress after a while. So, in vain, I looked at the solution. The ...
0
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1answer
94 views

Continued fraction of $\phi^3$

I found that $$\phi^3=4+\cfrac1{\small{4+\cfrac1{4+\cfrac1{4+\cfrac1{4+\ddots}}}}}$$ How should I prove this? Attempt: Suppose$$x= 4+\cfrac1{\small{4+\cfrac1{4+\cfrac1{4+\cfrac1{4+\ddots}}}}}$$ To ...
0
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1answer
46 views

Continued fractions approximation using golden ratio

Hello today my friend helped me with my problem, but he did not give me any additional informations why it works like that. Let's suppose that I need to get ln(n) using continued fractions. He told ...
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4answers
104 views

Does $a_{n}/a_{n-1}$ converge to the golden ratio for all Fibonacci-like sequences?

Yesterday a friend challenged me to prove that $$\lim_{n\rightarrow\infty}\frac{a_n}{a_{n-1}}=\varphi\; ,$$ where $\varphi$ is the golden ratio, for the Fibonacci series. I started rewriting the ...
0
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1answer
39 views

Proofs regarding the golden number

Given a succession $r(n)= 1 +\frac{1}{r(n-1)}$ where $r (1)=1$ and golden number $\phi =\frac{1+\sqrt{5}}{2}$. How do I prove that $$\left\lvert r(n)-\phi\right\rvert \leq \frac{1}{\phi ...
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0answers
26 views

Express solutions in terms of $\phi$

Previously, I solved the special transcendental equation $x=e^{t/\ln(x)}$. The solution is: $x=e^{-\sqrt{t}}$ for $0<x<1.$ One can define an equation: $e^{s/\ln(x)}=e^{t/\ln(1-x)},$ for $s,t \...
0
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1answer
57 views

More irrational than the Golden Ratio?

According to this video, $\varphi$ is the most irrational number, due to its continued fraction form having $1$, the smallest natural number, in the denominators. Is it not possible to construct a "...
13
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4answers
304 views

What is this function related with continued fractions?

Playing with continued fractions, I came with the idea of iterating the limit of the simplest one: $$1 + \cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cdots}}}}}\ = \Phi$$ And then I ...
5
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1answer
74 views

Ratio between farthest and second farthest distance

$n\geq 3$ points lie in three-dimensional space. What is the largest $c(n)$ such that there always exists a point for which the ratio between the distance to the farthest point from it and the ...
1
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2answers
70 views

Computing a Geometrical ratio $\frac{a}{b}$.

$XYZ$ is an equilateral triangle as shown on the image below. The aim is to find the ratio $\frac{a}{b}$. So far from the picture, it is easy to see that $b= \frac{YZ}{2}$. Does anyone have an ...
18
votes
1answer
260 views

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'...
0
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1answer
35 views

Ratio of 0's to 1's in the Fibonacci Word is the golden ratio

Define $S_0=0, S_1=01$. Then for $n\geq 2$ we define $S_n=S_{n-1}S_{n-2}$ (concatenating the previous sequence and the one before that). We obtain a limiting sequence, which we call the inifinite ...
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0answers
48 views

Is there a “natural” entire series associated to the Riemann zeta function whose radius of convergence is $\frac{1}{\sqrt{5}}$?

As a follow-up to Is there a hidden connection between RH and the golden ratio?, let's consider the plane $ P $ whose intersection with the Riemann sphere is the circle I denoted by $ \Gamma_{\Delta}...
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0answers
38 views

How does this drawing of a man's body in a pentagram suggest relationships to the golden ratio?

According to Wikipedia it does, but I can't see how. Wikipedia context: https://en.wikipedia.org/wiki/Golden_ratio#/media/File:Pentagram_and_human_body_(Agrippa).jpg
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3answers
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A pattern appearing in the powers of $\phi$

\begin{align} \phi^5 &= 11,\underline{0}901699\cdots\\ \phi^6 &= 17,\underline{9}44271\cdots\\ \phi^7 &= 29,\underline{6}34441\cdots\\ \phi^8 &= 46,\underline{9}7871\cdots\\ \phi^9 &...
0
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1answer
67 views

Show that $\int_0^{\infty} (\exp(x) - 2^{\sqrt 5 +1} + 1) (\coth(x) - 1) x^{\sqrt 5} dx = 0 $ [closed]

Show that $$ \int_0^{\infty} (\exp(x) - 2^{\sqrt 5 +1} + 1) (\coth(x) - 1) x^{\sqrt 5} dx = 0 $$ I wonder how many distinct methods there are.
1
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1answer
148 views

Prime number and golden ratio

I think the following is true, but can't show it. Let $p$ be a prime number, and let $f(x)$ be a polynomial which satisfies $${\Bigl(1-x+\frac{1}{x}\Bigr)}^p-1=f(x)+f\Bigl(-\frac{1}{x}\Bigr).$$ Then $...
0
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1answer
70 views

Golden and Silver ratios, how about the roots of $x^2+x-1=0$?

The golden and silver ratios are the roots of the equation $x^2-x-1=0$: $$\frac{1\pm\sqrt{5}}{2}.$$ They show up in the formula of Fibonacci numbers: $$F_n=\frac{1}{\sqrt{5}}\left(\frac{1+\sqrt{5}}{2}...
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3answers
74 views

Finding the exact value of $\sin 30°$ using golden ratio [closed]

$\sin \left(k\cdot 30^\circ\right) = \frac{\sqrt{2}}{4}\;\sqrt{\;4\;\pm_1\;\sqrt{\phi\,(a\phi+b\overline{\phi})}\;\pm_2\;\sqrt{\overline{\phi}\,(c\phi+d\overline{\phi})}\;}$ I'm currently trying to ...
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0answers
180 views

Golden Ratio in trigonometry

Assume that we have been asked to find the value of $\sin (18^\circ)$. We know that there are many ways to find it out. However, I'll be going with golden ratio! Let's draw a triangle whose apical ...
1
vote
1answer
87 views

Growth factor of the golden spiral

Wikipedia says, that a golden spiral is a logarithmic spiral whose growth factor is φ, the golden ratio. However, other sources[only in Czech, sorry] say that the "growth factor" of a spiral is ...
2
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1answer
49 views

How many ways can dominoes cover a $2 \times n$ rectangle? Justify proposed solution.

I was able to get that $d_n = d_{n-1}+d_{n-2}$ It isn't finished, because I have to solve this recursive equation. I read about Binet's formula, but I don't know the steps between this $$d_n = d_{n-...
6
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1answer
75 views

(Non-) Convergence of $\frac{1}{n} \sum_{k=0}^{n - 1} \exp\left(2i \pi [\frac{3 + \sqrt{5}}{2}]^k\right)$ when $n \to +\infty$

Let be $$\forall n > 0, S_n = \dfrac{1}{n} \sum\limits_{k=0}^{n - 1} \exp(2i\pi u_k),\quad \forall k \geq 0, u_k = \left(\dfrac{3 + \sqrt{5}}{2}\right)^k$$ I would like to prove or disprove the ...
6
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3answers
90 views

Matrix satisfying $A-I = A^{-1}$

Recall the (infinitely) continued fraction definition of the golden ratio \begin{align} \phi = 1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\cdots}}}}} \end{align} This is equivalent to ...
0
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1answer
89 views

Why is $x$ always a perfect square?

In the following equation: $$5 F_n^2 \pm 4 = x,$$ where $F_n$ is a Fibonacci number, and the $\pm 4$ shall be treated as $+4$ for even $n$ and $-4$ for odd $n$. Now, if the above requirements are ...
0
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1answer
155 views

Fibonacci numbers and perfect squares

Can this be simplified any further?: $\phi^{2n}$ + $\psi^{2n}$$-2(-1)^n$ $\pm4$ Where $\phi = (1+\sqrt5)/2 $ Where $\psi = (1-\sqrt5)/2 $ When n=even number, use +4 Whenn n=odd number, use -4 ...
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2answers
95 views

Powers of the golden ratio [closed]

Let $\phi$ be the golden ratio. I'm tasked to prove by other means than induction that $x$ in the next equation $$\phi^n =\phi F_n +x,$$ is actually a Fibonacci number. I have tried to apply Binet's ...
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0answers
58 views

Prove that $\operatorname{fib}(n)$ is the closest integer to $\frac{\phi^{n}}{\sqrt{5}}$

From SICP: Exercise 1.13: Prove that $\operatorname{fib}(n)$ is the closest integer to $$\frac{\phi^{n}}{\sqrt{5}}, \text{ where } \phi=\frac{1+\sqrt{5}}2.$$ Hint: Let $$\varphi=\frac{1-\sqrt{5}...
6
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1answer
94 views

On $3+\sqrt{11+\sqrt{11+\sqrt{11+\sqrt{11+\dots}}}}=\phi^4$ and friends

Let $\phi$ be the golden ratio. We know it has a beautiful infinite nested radical, $$\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\dots}}}}=\phi$$ However, it is also the case that, $$3+\sqrt{11+\sqrt{11+\sqrt{...
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3answers
46 views

Golden mean in this equation

Years ago I had started with this equation: $2^{1/3}=(R/2)(\sqrt{1+8/R^3}-1)$ And arrived at the result $2^{1/3}=\phi R$ Where phi, the golden ratio, is (sqrt(5) +1)/2. But at the moment can't ...
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0answers
130 views

$\int\limits_{0}^{t}(\exp{(\sqrt{x}-x^2)} )^ \text{erf(x)}\ \text{d}x,$ VS golden ratio For $t\geq 2$ ,

I have tried to get aother integral representation for the Golden ratio i have got the following representation . This integral is defined as : $$I(t)=\int\limits_{0}^{t}(\exp{(\sqrt{x}-x^2)} )^ \text{...
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1answer
300 views

Golden Rectangle into Golden Rectangles

Can these golden rectangles be rearranged to exactly cover the underlying cyan golden rectangle? That's the entire question. All that follows is related discussion. I want to make a more elegant ...
3
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1answer
99 views

Are $\varphi$ and $\varphi^3$ the only powers of $\varphi$ that are also metallic ratios?

Given the sequence generator function: $$F_\lambda(n+2)=\lambda F_\lambda(n+1)+F_\lambda(n);\quad F_\lambda(0)=0, F_\lambda(1)=1$$ where $\lambda=1$ corresponds to the Fibonacci sequence, $\lambda=2$ ...
2
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0answers
84 views

What is this “logarithmic golden ratio of scale-free networks” really called?

I have been working with "data science" related issues for almost a decade and I have always been huge fan of scale free networks and complexity theory. When I am building a domain rich model (...
3
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1answer
217 views

Golden ratio mod 1 distribution

If you plot the sequence $$x \leftarrow (x + \varphi)\ \mathrm{mod}\ 1$$ you get a nice scattering of numbers where no number is close to any previous number. This image shows the sequence after ...
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0answers
90 views

Geometric construction of golden angle

Is it possible to construct the golden angle using only a compass, ruler and pencil?
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337 views

The most complex formula for the golden ratio $\varphi$ that I have ever seen. How was it achieved?

I am fascinated by the following formula for the golden ratio $\varphi$: $$\Large\varphi = \frac{\sqrt{5}}{1 + \left(5^{3/4}\left(\frac{\sqrt{5} - 1}{2}\right)^{5/2} - 1\right)^{1/5}} - \frac{1}{e^{2\...
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0answers
82 views

Rational Trig Solutions for $n\ge 3$

Are there solutions to $$\sin(x+y)\sin(x-y)=n\ \sin(x)\sin(y)$$ for $n\ge 3$ where $x$ and $y$ are rational multiples of $\pi$? (excluding the trivial solutions when both sides are $0$). Known ...
4
votes
1answer
76 views

ireducible polynomials with coefficients in $\{0,-1\}$

I'm interested in the polynomials of the form $x^n -x^{n-1} - b_{n-2}x^{n-2} - \cdots -b_1 x -1$ with the coefficients $b_k$ being either zero or one. The prototype is of course the Golden mean $x^2-...
3
votes
1answer
101 views

Generalizing Odom's construction of the golden ratio

The artist and amateur mathematician George Odom found this nice construction for the golden ratio $\phi$ using an equilateral triangle and its circumcircle, $\hskip2.3in$ $\hskip3.3in$Fig. 1 Let $A$...
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0answers
64 views

Number theory in the quadratic field with golden section unit

I would like to ask a favor. Does anyone of you here have an access to the book 'Number theory in the quadratic field with golden section unit' by Fred Wayne Dodd? I just need to see Theorem 8.5 of ...
5
votes
0answers
212 views

Ford circles and the Fibonacci sequence

I graphed the Ford circles for the first few terms in the Fibonacci sequence $\frac{1}{1}, \frac{2}{1}, \frac{3}{2}, \ldots$ as well as a circle with radius $\frac{\sqrt{5}}{2}$ about the point $(\...
2
votes
1answer
85 views

Is $\frac{5\pi}{6}$ a transcendental or an algebraic number?

$\Phi^2$ is an algebraic number as it is the root of $x^2-3x+1=0$ So knowing that $\frac{\pi}{6} = \frac{\Phi^2}{5}$, which is a relation I saw, does it mean that $\frac{5\pi}{6}$ is algebraic ...
0
votes
1answer
77 views

A peculiar Diophantine equation

Solve the following equation in the variables $m\in\{1,2,\dots\}, n\in \{2,3,\dots \}$ and $k$ $\in\{1,\dots \lceil{ \frac{n}{2} } \rceil \}$ $$(m+\sqrt{m^2+4})^2(1-\cos\frac{2\pi}{n}) = 4(1-\cos\...