Related papers: How to Create a New Integer Sequence
We present an algorithm for effectively generating binary sequences which would be rated by people as highly likely to have been generated by a random process, such as flipping a fair coin.
We show that essentially the Fibonacci sequence is the unique binary recurrence which contains infinitely many three-term arithmetic progressions. A criterion for general linear recurrences having infinitely many three-term arithmetic…
The Fibonacci sequence is a series of positive integers in which, starting from $0$ and $1$, every number is the sum of two previous numbers, and the limiting ratio of any two consecutive numbers of this sequence is called the golden ratio.…
In this paper, we show that the concatenation of the Fibonacci sequence is \textit{normal} in base $10$, meaning every string of a given length, $k$, occurs as frequently as every other string of length $k$ (there are as many $1$'s as $2$'s…
Extending earlier work of R. Donaghey and P. J. Cameron, we investigate some canonical "eigen-sequences" associated with transformations of integer sequences. Several known sequences appear in a new setting: for instance the sequences (such…
Sequences of Genocchi numbers of the first and second kind are considered. For these numbers, an approach based on their representation using sequences of polynomials is developed. Based on this approach, for these numbers some identities…
This paper gives a brief description of the author's database of integer sequences, now over 35 years old, together with a selection of a few of the most interesting sequences in the table. Many unsolved problems are mentioned.
Let (F_n^{(k)})_{n\geq -(k-2)} be the k-generalized Fibonacci sequence, defined as the linear recurrence sequence whose first k terms are \(0, 0, \ldots, 0, 1\), and whose subsequent terms are determined by the sum of the preceding k terms.…
The classical Fibonacci sequence is known to exhibit many fascinating properties. In this paper, we explore the Fibonacci sequence and integer sequences generated by second order linear recurrence relations with positive integer…
In this paper, we define a variant of Fibonacci-like sequences that we call prime Fibonacci sequences, where one takes the sum of the previous two terms and returns the smallest odd prime divisor of that sum as the next term. We prove that…
We define a sequence of positive integers recursively, where each term is determined as follows: starting with a given positive integer, if the term is odd, the next is the sum of its positive divisors; if the term is even, the subsequent…
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…
D. S. Hong and P. Pongsriiam have provided a necessary and sufficient condition for the generating function for Fibonacci numbers (resp. the Lucas numbers) to be an integer value, for rational numbers. In other words, their results relate…
In this short article, we study different problems described as initial value problems of discrete differential equations and develop a a transform method called the sigma transform, a discrete version of the continuous Laplace transform to…
The Tribonacci sequence is a well-known example of third order recurrence sequence, which belongs to a particular class of recursive sequences. In this article, other generalized Tribonacci sequence is introduced and defined by…
In this paper we study some structure properties of primitive weird numbers in terms of their factorization. We give sufficient conditions to ensure that a positive integer is weird. Two algorithms for generating weird numbers having a…
Let $B$ be a finite set of natural numbers or complex numbers. Product set corresponding to $B$ is defined by $B.B:=\{ab:a,b\in B\}$. In this paper we give an upper bound for longest length of consecutive terms of a polynomial sequence…
A construction sequence for a graph is a listing of the elements of the graph (the set of vertices and edges) such that each edge follows both its endpoints. The construction number of the graph is the number of such sequences. We determine…
Researchers have demonstrated that humans are unable to generate a sequence of random numbers that corresponds in a statistical sense to a simple distribution such as the uniform distribution. The purpose of this article is to present the…
An elementary method for computing various prime sequences using the sequence of Farey sequences is described.