Related papers: Eight Hateful Sequences
Let $N$ be a positive integer. A sequence $X=(x_1,x_2,\ldots,x_N)$ of points in the unit interval $[0,1)$ is piercing if $\{x_1,x_2,\ldots,x_n\}\cap \left[\frac{i}{n},\frac{i+1}{n} \right) \neq\emptyset$ holds for every $n=1,2,\ldots, N$…
We present a certain generalization of a recent result of M. I. Cirnu on linear recurrence relations with coefficient in progressions [2]. We provide some interesting examples related to some well-known integer sequences, such as Fibonacci…
Consider the celebrated Lyness recurrence $x_{n+2}=(a+x_{n+1})/x_{n}$ with $a\in\Q$. First we prove that there exist initial conditions and values of $a$ for which it generates periodic sequences of rational numbers with prime periods…
A collection $B$ of patterns is called inversion monotone if $\mathrm{av}_n^k(B)$, the number of $B$-avoiding permutations of length $n$ with $k$ inversions, is weakly increasing in $n$ for any fixed $k$. In 2012, Claesson, Jel\'inek and…
We generalise our earlier work on the number of squares in binary recurrence sequences, $\left\{ y_{k} \right\}_{k \geq -\infty}$. In the notation of our previous papers, here we consider the case when $N_{\alpha}$ is any negative integer…
A $d$-subsequence of a sequence $\varphi = x_1\dots x_n$ is a subsequence $x_i x_{i+d} x_{i+2d} \dots$, for any positive integer $d$ and any $i$, $1 \le i \le n$. A \textit{$k$-Thue sequence} is a sequence in which every $d$-subsequence,…
Rosenfeld in 1974 conjectured that there is an integer N > 8 such that every tournament of order n > N contains every non-directed cycle of order n. We prove that, with exactly 35 exceptions, every tournament of order n > 2 contains each…
The $3x+1$ Problem asks if whether for every natural number $n$, there exists a finite number of iterations of the piecewise function $$f(2n)=n, \quad f(2n-1)=6n-2, $$ with an iterate equal to the number $1$, or in other words, every…
Let $\mathscr{X}$ be the set of positive real sequences $x=(x_n)$ such that the series $\sum_n x_n$ is divergent. For each $x \in \mathscr{X}$, let $\mathcal{I}_x$ be the collection of all $A\subseteq \mathbf{N}$ such that the subseries…
Given a set of integers containing no 3-term arithmetic progressions, one constructs a Stanley sequence by choosing integers greedily without forming such a progression. Independent Stanley sequences are a "well-structured" class of Stanley…
By following the same construction pattern which Martin Davis proposed in a 1968 paper of his, we have obtained six quaternary quartic Diophantine equations that candidate as `rule-them-all' equations: proving that one of them has only a…
In 1999, Neil Calkin and Herbert Wilf wrote "Recounting the rationals" which gave an explicit bijection between the positive integers and the positive rationals. We find several different (some new) ways to construct this enumeration and…
The concept of antimagic labelings of a graph is to produce distinct vertex sums by labeling edges through consecutive numbers starting from one. A long-standing conjecture is that every connected graph, except a single edge, is antimagic.…
Very odd sequences were introduced in 1973 by J. Pelikan who conjectured that there were none of length >=5. This conjecture was disproved by MacWilliams and Odlyzko in 1977 who proved there are in fact many very odd sequences. We give…
The first author recently introduced an integer sequence now numbered A355519 in OEIS. This sequence arose from counting bracket tournaments; its study evokes the analysis of the Catalan triangle (sequence A009766 in OEIS) and the related…
In this article, we reduce the unsolved problem of convergence of Collatz sequences to convergence of Collatz sequences of odd numbers that are divisible by 3. We give an elementary proof of the fact that a Collatz sequence does not…
Consider the number of permutations in the symmetric group on n letters that contain c copies of a given pattern. As c varies (with n held fixed) these numbers form a sequence whose properties we study for the monotone patterns and the…
We count the number of alignments of $N \ge 1$ sequences when match-up types are from a specified set $S\subseteq \mathbb{N}^N$. Equivalently, we count the number of nonnegative integer matrices whose rows sum to a given fixed vector and…
The comma sequence (1, 12, 35, 94, ...) is the lexicographically earliest sequence such that the difference of consecutive terms equals the concatenation of the digits on either side of the comma separating them. The behavior of a…
Existing computational models to understand hate speech typically frame the problem as a simple classification task, bypassing the understanding of hate symbols (e.g., 14 words, kigy) and their secret connotations. In this paper, we propose…