Related papers: How to Create a New Integer Sequence
The research aims to construct a new type of matrix called the Fibonacci-Hessenberg-Lorentz matrix by multiplying Fibonacci-Hessenberg matrices with Lorentz matrix multiplication. The study will start by examining the properties of…
We derive several identities for arbitrary homogeneous second order recurrence sequences with constant coefficients. The results are then applied to present a unified study of six well known integer sequences, namely the Fibonacci sequence,…
The purpose of this paper is twofold. First, the definition of new statistical convergence with Fibonacci sequence is given and some fundamental properties of statistical convergence are examined. Second, approximation theory worked as a…
A \Def{composition} of a positive integer $n$ is a $k$-tuple $(\l_1, \l_2, \dots, \l_k) \in \Z_{> 0}^k$ such that $n = \l_1 + \l_2 + \dots + \l_k$. Our goal is to enumerate those compositions whose parts $\l_1, \l_2, \dots, \l_k$ avoid a…
Given a real number $0.a_1a_2 a_3\dots$ that is normal to base $b$, we examine increasing sequences $n_i$ so that the number $0.a_{n_1}a_{n_2}a_{n_3}\dots$ are normal to base $b$. Classically it is known that if the $n_i$ form an arithmetic…
Infinite sequences are of tremendous theoretical and practical importance, and in the Information Age sequences of 0s and 1s are of particular interest. Over the past century, the field of symbolic dynamics has developed to study sequences…
There are scattered results in the literature showing that the leading principal minors of certain infinite integer matrices form the Fibonacci and Lucas sequences. In this article, among other results, we have obtained new families of…
This tutorial provides a comprehensive and in-depth view of the research on procedures, primarily in Natural Language Processing. A procedure is a sequence of steps intended to achieve some goal. Understanding procedures in natural language…
We speculate on the distribution of primes in exponentially growing, linear recurrence sequences $(u_n)_{n\geq 0}$ in the integers. By tweaking a heuristic which is successfully used to predict the number of prime values of polynomials, we…
In this paper, we define the bi-periodic Fibonacci matrix sequence that represent bi-periodic Fibonacci numbers. Then, we investigate generating function, Binet formula and summations of bi-periodic Fibonacci matrix sequence. After that, we…
The Fibonacci sequence is obtained as weighted sum along the rows in the Pascal triangle by choosing a periodic up-and-down pattern of weights from the set $\{-1,-\frac{1}{2},0, \frac{1}{2}, 1\}$. A graphical illustration of this identity…
We present a self-learning approach for synthesizing programs from integer sequences. Our method relies on a tree search guided by a learned policy. Our system is tested on the On-Line Encyclopedia of Integer Sequences. There, it discovers,…
Farey sequences, Stern-Brocot sequences, the Calkin-Wilf sequences are shown to be generated via almost identical second order recurrence relations. These sequences have combinatorial, computational, and geometric applications, and are…
Given any two sequences of complex numbers, we establish simple relations between their binomial convolution and the binomial convolution of their individual binomial transforms. We employ these relations to derive new identities involving…
Fibonacci sequence, generated by summing the preceding two terms, is a classical sequence renowned for its elegant properties. In this paper, leveraging properties of generalized Fibonacci sequences and formulas for consecutive sums of…
The chronicle of prime numbers travel back thousands of years in human history. Not only the traits of prime numbers have surprised people, but also all those endeavors made for ages to find a pattern in the appearance of prime numbers has…
It's the age-old recurrence with a twist: sum the last two terms and if the result is composite, divide by its smallest prime divisor to get the next term (e.g., 0, 1, 1, 2, 3, 5, 4, 3, 7, ...). These sequences exhibit pseudo-random…
We present a particularly nice Binet-style formula that can be used to produce the k-generalized Fibonacci numbers (that is, the Tribonaccis, Tetranaccis, etc). Furthermore, we show that in fact one needs only take the integer closest to…
In array-based DNA synthesis, multiple strands of DNA are synthesized in parallel to reduce the time cost from the sum of their lengths to the length their shortest common supersequences. To maximize the amount of information that can be…
We define impulse response sequence in the set of all linear recurring sequences satisfying a linear recurrence relation of order $r$. The generating function and expression of the impulse response sequence are presented. Some identities of…