All Questions
Tagged with co.combinatorics ds.dynamical-systems
13 questions
16
votes
6
answers
1k
views
A Leibniz-like formula for $(f(x) \frac{d}{dx})^n f(x)$?
Let $f(x)$ be sufficiently regular (e.g. a smooth function or a formal power series in characteristic 0 etc.). In my research the following recursion made a surprising entrance
$$
f_1(x) = f(x),\ f_{n+...
- 7,127
57
votes
0
answers
3k
views
On the first sequence without triple in arithmetic progression
In this Numberphile video (from 3:36 to 7:41), Neil Sloane explains an amazing sequence:
It is the lexicographically first among the sequences of positive integers without triple in arithmetic ...
31
votes
1
answer
1k
views
Vanishing line on Conway's game of life
If the initial state of Conway's game of life is a line of $n \in [0,100]$ alive cells, then it vanishes completely after some steps iff $n \in \{0,1,2,6,14,15,18,19,23,24 \}$. See below for $n=24$.
...
23
votes
1
answer
2k
views
Time for Langton's ant to cover a "square" torus
Langton's ant is a cellular automaton running as follows:
Squares on a plane are colored variously either black or white. We
arbitrarily identify one square as the "ant". The ant can travel in
...
22
votes
3
answers
1k
views
Cyclic action on Kreweras walks
A Kreweras walk of length $3n$ is a word consisting of $n$ $A$'s, $n$ $B$'s, and $n$ $C$'s such that in any prefix there are at least as many $A$'s as $B$'s, and at least as many $A$'s as $C$'s. For ...
- 24.2k
16
votes
3
answers
2k
views
Periodic orbits and polynomials
There are two simple and classic enumerations that still I'm puzzled about. Let's start with a simple counting problem from a well-known dynamical system.
fact 1 Consider the "tent map" f:[0,1]→[...
- 60.5k
15
votes
1
answer
1k
views
In how many steps a random walk visits all the elements of a finite group, with a probability 1/2?
This question is a variation of the return to the origin problem.
Let $G$ be the finite group $\mathbb{Z}/n \times \mathbb{Z}/n$ and let the random transformation $T: G \to G$ such that $T(a,b) = (...
13
votes
1
answer
597
views
Does Langton's ant cover every n by 6 gridded torus?
This post follows this other post about times cover by Langton's ant of $n$ by $n$ gridded torus.
For $n$ by $n$ gridded torus, I've checked for $n \le 1000$ that the ant covers all. This fact needs ...
10
votes
1
answer
673
views
Given any finite relation $R$ what is the cardinality of $\langle R\rangle=\{\underbrace{R\circ R\cdots \circ R}_{n\text{ times}}:n\in\mathbb{N}\}$?
Given any finite relation $R$ if we let $\circ$ denote relation composition and define $R^n=\underbrace{R\circ R\cdots \circ R}_{n\text{ times}}$ then does there exist an explicit formula for the ...
- 2,506
8
votes
2
answers
2k
views
5n+1 sequence starting at 7
Consider the following variant of the Collatz function: $f:\mathbb N\rightarrow\mathbb N$ is defined by
\begin{equation}
f(n):=\begin{cases}
n/2 & \text{if $n$ is even}\\
5n+1 & \...
- 654
8
votes
1
answer
436
views
The graph of Rule 110 and vertices degree
Consider the elementary cellular automaton called Rule 110 (famous for being Turing complete):
It induces a map $R: \mathbb{N} \to \mathbb{N}$ such that the binary representation of $R(n)$ is ...
7
votes
4
answers
1k
views
The non-convergence of f(f(x))=exp(x)-1 and labeled rooted trees
This question is closely related to MO f(f(x))=exp(x)-1 and other functions “just in the middle” between linear and exponential. Consider $e^{e^x-1}$, this is the generating function of the Bell ...
user37691
2
votes
3
answers
638
views
The critical exponent function
It is a known fact [1] that, for every $c\in (1,\infty]$, it is possible to find a finite alphabet $\mathcal{A}$ and a word $w\in \mathcal{A}^\omega$ such that $w$ has critical exponent $c$. It looks ...
- 4,459