Related papers: Anti-recurrence sequences
We define a family of meta-Fibonacci sequences where the order of the of recursion at stage n is a variable r(n), and the n^{th} term of a sequence is the sum of the previous r(n) terms. For the terms of any such sequence, we give upper and…
Collatz Conjecture sequences increase and decrease in seemingly random fashion. By identifying and analyzing the forms of numbers, we discover that Collatz sequences are governed by very specific, well-defined rules, which we call cascades.
Let $(X_{k})_{k\geq 1}$ and $(Y_k)_{k\geq 1}$ be the sequence of $X$ and $Y$-coordinates of the positive integer solutions $(x, y)$ of the equation $x^2 - dy^2 = t$. In this paper we completely describe those recurrence sequences such that…
To illustrate that the notion of convergence of submodular function sequences fits reasonably into the limit theory of graphs, we describe several classes of matroids and other submodular setfunctions for which convergence of appropriate…
Given the congruence lattice L of a finite algebra A with a Mal'cev term, we look for those sequences of operations on L that are sequences of higher commutator operations of expansions of A. The properties of higher commutators proved so…
We establish a supercongruence conjectured by Almkvist and Zudilin, by proving a corresponding $q$-supercongruence. Similar $q$-supercongruences are established for binomial coefficients and the Ap\'{e}ry numbers, by means of a general…
Recurrence plots provide a graphical representation of the recurrent patterns in a timeseries, the quantification of which is a relatively new field. Here we derive analytical expressions which relate the values of key statistics, notably…
We consider real sequences $(f_n)$ that satisfy a linear recurrence with constant coefficients. We show that the density of the positivity set of such a sequence always exists. In the special case where the sequence has no positive…
We review how the monotone pattern compares to other patterns in terms of enumerative results on pattern avoiding permutations. We consider three natural definitions of pattern avoidance, give an overview of classic and recent formulas, and…
This study involves definitions for multiple-counting regular and summation sequences of rho. My paper introduces and proves recurrent relationships for multiple-counting sequences and shows their association with Fermat's little theorem. I…
Recently, Z. W. Sun put forward a series of conjectures on monotonicity of combinatorial sequences in the form of $\{z_n/z_{n-1}\}_{n=N}^\infty$ and $\{\sqrt[n+1]{z_{n+1}}/\sqrt[n]{z_n}\}_{n=N}^\infty$ for some positive integer $N$, where…
This paper presents a reinterpretation of a second-order linear recurrence sequence as a sequence of continuants derived from the convergents to a continued fraction. As a result, we are able to derive the generating function and Binet…
The study of pinnacle sets has been a recent area of interest in combinatorics. Given a permutation, its pinnacle set is the set of all values larger than the values on either side of it. Largely inspired by conjectures posed by Davis,…
The Thue--Morse sequence is a prototypical automatic sequence found in diverse areas of mathematics, and in computer science. We study occurrences of factors $w$ within this sequence, more precisely, the sequence of gaps between consecutive…
Motivated by a problem in quantum field theory, we study the up and down structure of circular and linear permutations. In particular, we count the length of the (alternating) runs of permutations by representing them as monomials and find…
We study generalized sums of linear orders. These are binary operations that, given linear orders $A$ and $B$, return an order $A \oplus B$ that can be decomposed as an isomorphic copy of $A$ interleaved with a copy of $B$. We show that…
A sum-and-distance system is a collection of finite sets of integers such that the sums and differences formed by taking one element from each set generate a prescribed arithmetic progression. Such systems, with two component sets, arise…
Many automatic sequences, such as the Thue-Morse sequence or the Rudin-Shapiro sequence, have some desirable features of pseudorandomness such as a large linear complexity and a small well-distribution measure. However, they also have some…
We consider quivers/skew-symmetric matrices under the action of mutation (in the cluster algebra sense). We classify those which are isomorphic to their own mutation via a cycle permuting all the vertices, and give families of quivers which…
We present in this paper a new method to deal with automatic sequences. This method allows us to prove a M\"obius-randomness-principle for automatic sequences from which we deduce the Sarnak conjecture for this class of sequences.…