Related papers: Eight Hateful Sequences
In this expository article we collect the integer sequences that count several different types of matrices over finite fields and provide references to the Online Encyclopedia of Integer Sequences (OEIS). Section 1 contains the sequences,…
Throughout history, recreational mathematics has always played a prominent role in advancing research. Following in this tradition, in this paper we extend some recent work with crazy sequential representations of numbers- equations made of…
Fifteen years ago, then-Carleton-undergrad Isaac Hodes, proved that the Golden Ratio is evil. In this modest contribution to human knowledge, we show that in fact, every fifth real number is evil, and we present lots of other interesting…
We study additive properties of consecutive prime numbers and the primality of the sums they generate. For a given prime number $p_n$, we consider the sums \[ S_k(p_n) = p_n + p_{n+1} + \cdots + p_{n+k-1}, \] where $k \ge 3$ is an odd…
We resolve a $1000 Erd\H{o}s prize problem, complete with formal verification generated by a large language model. In over a dozen papers, beginning in 1976 and spanning two decades, Paul Erd\H{o}s repeatedly posed one of his "favourite"…
We study the problem of generating interesting integer sequences with a combinatorial interpretation. For this we introduce a two-step approach. In the first step, we generate first-order logic sentences which define some combinatorial…
One of the most classical results in Ramsey theory is the theorem of Erd\H{o}s and Szekeres from 1935, which says that every sequence of more than $k^2$ numbers contains a monotone subsequence of length $k+1$. We address the following…
First we reprove two results in additive number theory due to Dombi and Chen & Wang, respectively, on the number of representations of n as the sum of two odious or evil numbers, using techniques from automata theory and logic. We also use…
`Terquem's problem' is a name given in the twentieth century to the problem of enumerating certain integer sequences whose entries alternate in parity. In particular, this problem asks for the count of strictly increasing length $m$…
Odlyzko and Stanley introduced a greedy algorithm for constructing infinite sequences with no 3-term arithmetic progressions when beginning with a finite set with no 3-term arithmetic progressions. The sequences constructed from this…
Markov numbers are integers that appear in the solution triples of the Diophantine equation, $x^2+y^2+z^2=3xyz$, called the Markov equation. A classical topic in number theory, these numbers are related to many areas of mathematics such as…
An inversion sequence of length $n$ is an integer sequence $e=e_{1}e_{2}\dots e_{n}$ such that $0\leq e_{i}<i$ for each $i$. Corteel--Martinez--Savage--Weselcouch and Mansour--Shattuck began the study of patterns in inversion sequences,…
The first author introduced a sequence of polynomials (\cite{8}, sequence A174531) defined recursively. One of the main results of this study is proof of the integrality of its coefficients.
Labeled infinite trees provide combinatorial interpretations for many integer sequences generated by nested recurrence relations. Typically, such sequences are monotone increasing. Several of these sequences also have straightforward…
It is known that the following five counting problems lead to the same integer sequence~$f_t(n)$: the number of nonequivalent compact Huffman codes of length~$n$ over an alphabet of $t$ letters, the number of `nonequivalent' canonical…
In [Kit1] Kitaev discussed simultaneous avoidance of two 3-patterns with no internal dashes, that is, where the patterns correspond to contiguous subwords in a permutation. In three essentially different cases, the numbers of such…
This paper is a supplement to a talk for mathematics teachers given at the 2016 LSU Mathematics Contest for High School Students. The paper covers more details and aspects than could be covered in the talk. We start with an interesting…
Let $L$ be a primitive Gaussian line, that is, a line in the complex plane that contains two, and hence infinitely many, coprime Gaussian integers. We prove that there exists an integer $G_L$ such that for every integer $n\geq G_L$ there…
Given a finite nonempty sequence S of integers, write it as XY^k, where Y^k is a power of greatest exponent that is a suffix of S: this k is the curling number of S. The Curling Number Conjecture is that if one starts with any initial…
A sequence $S=s_{1}s_{2}..._{n}$ is \emph{nonrepetitive} if no two adjacent blocks of $S$ are identical. In 1906 Thue proved that there exist arbitrarily long nonrepetitive sequences over 3-element set of symbols. We study a generalization…