Related papers: A Recreational Application of Two Integer Sequence…
In 2012, Diem introduced a new figure of merit for cryptographic sequences called expansion complexity. In this paper, we slightly modify this notion to obtain the so-called irreducible-expansion complexity which is more suitable for…
Bond percolation on Cayley graphs provides examples of random graphs. Other examples arise from the dynamical study of proper repetitive subgraphs of Cayley graphs. In this paper we demonstrate that these two families have mutually singular…
We investigate the special class of formulas made up of arbitrary but finite com- binations of addition, multiplication, and exponentiation gates. The inputs to these formulas are restricted to the integral unit 1. In connection with such…
The classic string indexing problem is to preprocess a string S into a compact data structure that supports efficient pattern matching queries. Typical queries include existential queries (decide if the pattern occurs in S), reporting…
We study a class of combinatorial objects that we call "decorated trees". These consist of vertices, arrows and edges, where each edge is decorated by two integers (one near each of its endpoints), each arrow is decorated by an integer, and…
A Sidon sequence is a sequence of integers a_1 < a_2 < a_3 < ... with the property that the sums a_i+a_j (i\le j) are distinct. This work contains a survey of Sidon sequences and their generalizations, and an extensive annotated and…
I propose and investigate the use of continuous functional equations for the study of meta-Fibonacci integer sequences. This exploratory study includes three sequences with quite different behavior: Conway's famous sequence $A(n)=…
We discuss properties of integers in base 3/2. We also introduce many new sequences related to base 3/2. Some sequences discuss patterns related to integers in base 3/2. Other sequence are analogues of famous base-10 sequences: we discuss…
We prove an infinite family of lacunary recurrences for the Lucas numbers using combinatorial means.
We give an elementary proof of an interesting combinatorial identity which is of particular interest in graph theory and its applications. Two applications to enumeration of forests with closed-form expressions are given.
A generalized Beatty sequence is a sequence $V$ defined by $V(n)=p\lfloor{n\alpha}\rfloor+qn +r$, for $n=1,2,\dots$, where $\alpha$ is a real number, and $p,q,r$ are integers. These occur in several problems, as for instance in homomorphic…
In this paper we generalize an identity first given by Heinrich Eduard Heine in his treatise, {\it Handbuch der Kugelfunctionen, Theorie und Anwendungen (1881), which gives a Fourier series for $1/[z-\cos\psi]^{1/2}$, for $z,\psi\in\R$, and…
A 1-dimensional generalization of the well known Logistic Map is proposed. The proposed family of maps is referred to as the B-Exponential Map. The dynamics of this map are analyzed and found to have interesting properties. In particular,…
Given a sequence 1, 1, 5, 23, 135, 925, 7285, 64755, 641075, 6993545, 83339745,..., how can we guess a formula for it? This article will quickly walk you through the concept of ansatz for classes of polynomial, $C$-finite, holonomic, and…
The starting point of this work is an equality between two quantities $A$ and $B$ found in the literature, which involve the {\em doubling-modulo-an-odd-integer} map, i.e., $x\in {\mathbb N} \mapsto 2x \bmod{(2n+1)}$ for some positive…
We consider sequences of the type $A_n=6A_{n-1}-A_{n-2}, A_0=r, A_1=s$ ($r$ and $s$ integers) and show that all sequences that solve particular cases of the Pell generalized equation are expressible as a constant times one of four…
We study sets of integers that can be defined by the vanishing of a generalised polynomial expression. We show that this includes sets of values of linear recurrent sequences of Salem type and some linear recurrent sequences of Pisot type.…
We study the integer sequence v_n of numbers of lines in hypersurfaces of degree 2n-3 of P^n, n>1. We prove a number of congruence properties of these numbers of several different types. Furthermore, the asymptotics of the v_n are described…
For each positive integer n greater than or equal to 2, a new approach to expressing real numbers as sequences of nonnegative integers is given. The n=2 case is equivalent to the standard continued fraction algorithm. For n=3, it reduces to…
In the note, the authors give a unified proof of Identities~67, 84, and~85 in the monograph "M. Z. Spivey, The Art of Proving Binomial Identities, Discrete Mathematics and its Applications, CRC Press, Boca Raton, FL, 2019; available online…