0
votes
2answers
57 views

How to interpret the OEIS function for the “even fractal sequence” A103391 (1, 2, 2, 3, 2, 4, 3, 5, …)

I'm interested in a particular integer sequence that is the same as itself when you remove all of the even-indexed members of the sequence. It begins (1, 2, 2, 3, 2, 4, 3, 5, ...). I looked it up in ...
0
votes
1answer
87 views

Can someone help me find the sum of the following series?

I am working on one of the fractals and finding its convergent area. $$\begin{align} S & = 1+3\left(\frac{1}{9}+4(\frac{1}{9^2})+4^2(\frac{1}{9^3})+...\right)\\ & = 1+3*\sum_{i=0}^{\infty} ...
4
votes
3answers
244 views

Discuss the convergence of $ \left \{ a_n \right\} $ where $ a_{n+1}=\frac{a_0}{2}+\frac{a_n^2}{2},n\geq 1 $

Let $$ a_{n+1} = \dfrac{a_0}{2} + \dfrac{a_n^2}{2} $$ where $ a_1 = \dfrac{a_0}{2} $ and $ n\geq 1 $ Discuss the convergence of $ \left\{a_n\right\} $
4
votes
0answers
95 views

The Tribonacci constant and the Dragon

Let $x = \frac{\ln T}{\ln 2} = 0.879146\dots$ where $T$ is the tribonacci constant, then x solves the transcendental equation, $$4^x(2^x-1)=(2^x+1)$$ Let $x = \frac{\ln y}{\ln 2} = 1.523627\dots$ ...
4
votes
1answer
125 views

fractal structure of the sum of squares function

The sum of squares function came up at a job interview, corrected for signs and symmetry. $d_2(n)=\#\{(x,y): x^2 + y^2 = n\}$ However, want $(x,y)\sim (\pm x, \pm y) \sim (y,x)$. The first ...
6
votes
3answers
1k views

Is Fractal perimeter always infinite?

Looking for information on fractals through google I have read several time that one characteristic of fractals is : finite area infinite perimeter Although I can feel the area is finite (at ...
8
votes
2answers
280 views

How to prove a property regarding periodicities of points in the Mandelbrot set?

While studying a visual representation the Mandelbrot set, I have come across a very interesting property: For any point inside the same primary bulb (a circular-like 'decoration' attached to the ...
8
votes
1answer
260 views

The Mandelbrot Set Membership

To define the Mandelbrot Set we consider a sequence of complex numbers $z_0$, $z_1$, $z_2$, $z_3$, with the following conditions: $$ \begin{cases} z_{n+1} &= &z_n^2 + c &\text{ for }n\geq ...
-1
votes
1answer
182 views

How to find the area. Linked with another question. [duplicate]

Possible Duplicate: Is value of $\pi = 4$? In this question we discussed why the fake proof is wrong. But, what about the area? The process converges to the same area of the circle ...
9
votes
1answer
389 views

Number of limit points of a continued exponential

Inspired by the work of C. Bender, I recently played with continued exponentials (like continued fractions but with exponential functions ;) ). Given all prefactors are equal to 1, the continued ...
0
votes
2answers
282 views

How do I plot this red dot?

Here is the dot And here is how I arrived at it ยป Pretty simple IFS using a 2 x 2 grid as the base for the iterations. Is there a way to describe this point as well as its siblings, all of which ...