Related papers: Turyn-type sequences: Classification, Enumeration …

The generalized Tur\'an number $\text{ex}(n,H,\mathcal{F})$ denotes the maximum number of copies of $H$ in an $n$-vertex graph which contains no copies of any graph in a family $\mathcal{F}$ of graphs. The generalized rational exponents…

SU(4) Einstein-Yang-Mills theory possesses sequences of static spherically symmetric globally regular and black hole solutions. Considering solutions with a purely magnetic gauge field, based on the 4-dimensional embedding of $su(2)$ in…

The generalized Tur\'an number $\mathrm{ex}(n, H, \mathcal{F})$ is defined as the maximum number of copies of a graph $H$ in an $n$-vertex graph that does not contain any graph $F \in \mathcal{F}$. Alon and Frankl initiated the study of…

The Tribonacci sequence $\mathbb{T}$ is the fixed point of the substitution $\sigma(a,b,c)=(ab,ac,a)$. In this note, we give the explicit expressions of the numbers of distinct squares and cubes in $\mathbb{T}[1,n]$ (the prefix of…

It is well-known that any sequence of at least N integers contains a subsequence whose sum is 0 (mod N). However, there can be very few subsequences with this property (e.g. if the initial sequence is just N 1's, then there is only one…

For graphs $H$ and $F$, the generalized Tur\'an number $ex(n,H,F)$ is the largest number of copies of $H$ in an $F$-free graph on $n$ vertices. We consider this problem when both $H$ and $F$ have at most four vertices. We give sharp results…

Golomb's sequence is the unique nondecreasing sequence of positive integers in which each $n$ appears exactly $a(n)$ times. It satisfies the global self-referential rule \[ a\bigl(a(n)+a(n-1)+\cdots+a(1)\bigr)=n, \] grows smoothly like a…

Let T(m,n) denote the number of ways to tile an m-by-n rectangle with dominos. For any fixed m, the numbers T(m,n) satisfy a linear recurrence relation, and so may be extrapolated to negative values of n; these extrapolated values satisfy…

Borwein and Mossinghoff investigated the Rudin-Shapiro-like sequences, which are infinite families of binary sequences, usually represented as polynomials. Each family of Rudin-Shapiro-like sequences is obtained from a starting sequence…

This paper is a continuation of previous work of the authors. We extend one of the theorems that gave a way to construct equilateral triangles whose vertices have integer coordinates to the general situation. An approximate extrapolation…

We consider the problem of counting the occurrences of patterns of the form $xy-z$ within flattened permutations of a given length. Using symmetric functions, we find recurrence relations satisfied by the distributions on $\mathcal{S}_n$…

Given a family $\mathcal{F}$ of $r$-graphs, the Tur\'{a}n number of $\mathcal{F}$ for a given positive integer $N$, denoted by $ex(N,\mathcal{F})$, is the maximum number of edges of an $r$-graph on $N$ vertices that does not contain any…

A tuple $(T_1,\dots,T_n)$ of continuous linear operators on a topological vector space $X$ is called hypercyclic if there is $x\in X$ such that the the orbit of $x$ under the action of the semigroup generated by $T_1,\dots,T_n$ is dense in…

We study the recursions $A(n) = A(n-a-A^k(n-b)) + A(A^k(n-b))$ where $a \geq 0$, $b \geq 1$ are integers and the superscript $k$ denotes a $k$-fold composition, and also the recursion $C(n) = C(n-s-C(n-1)) + C(n-s-2-C(n-3))$ where $s \geq…

The Turing degree spectrum of a countable structure $\mathcal{A}$ is the set of all Turing degrees of isomorphic copies of $\mathcal{A}$. The Turing degree of the isomorphism type of $\mathcal{A}$, if it exists, is the least Turing degree…

In this paper we use the Recursion Theorem to show the existence of various infinite sequences and sets. Our main result is that there is an increasing sequence e_0, e_1, e_2 .. such that W_{e_n}={e_{n+1}} for every n. Similarly, we prove…

In this article we prove that all completely multiplicative automatic sequences $(a_n)_{n \in \mathbf{N}}$ defined on $\mathbf{C}$, vanishing or not, can be written in the form $a_n=b_n\chi_n$ where $(b_n)_{n \in \mathbf{N}}$ is an almost…

Generalized Fibonacci-like sequences appear in finite difference approximations of the Partial Differential Equations based upon replacing partial differential equations by finite difference equations. This paper studies properties of the…

For positive integers $k,n$, a de Bruijn sequence $B(k,n)$ is a finite sequence of elements drawn from $k$ characters whose subwords of length $n$ are exactly the $k^n$ words of length $n$ on $k$ characters. This paper introduces the…

We study the triangular array defined by the Graham--Knuth--Patashnik recurrence $T(n,k) = (\alpha n + \beta k + \gamma)\, T(n-1,k)+(\alpha' n + \beta' k + \gamma') \, T(n-1,k-1)$ with initial condition $T(0,k) = \delta_{k0}$ and parameters…