Sequence Of Integers Research Articles

Integral values of generating functions of recursive sequences

Suppose that a0,a1,… is an integer sequence which satisfies a recurrence relation with constant coefficients, and let T(x)=f(x)/g(x) be its generating function, where f(x) and g(x) have no common factors in Z[x]. In this article, we study the problem of finding the rational values of x such that T(x) is an integer. We say that such a number is good for the sequence. Our first main result is that if g(x) has at least two different irreducible factors, or if g(x) has a single irreducible factor of degree at least 3, then the sequence has only finitely many good values. We also study sequences of the form 0,1,… for which the recurrence relation has order 2. Among other results, we show that under a mild condition on the recurrence relation, the sequence has infinitely many good values, and we give a constructive method to find all of them.

The Ruler Sequence Revisited: A Dynamic Perspective

The Ruler function or the Gros sequence is a classical infinite integer sequence that underlies some interesting mathematical problems. In this paper, we provide four new problems containing this type of sequence: (i) demographic discrete dynamical automaton, (ii) the middle interval Cantor set, (iii) construction by duplication of polygons and (iv) the horizontal visibility sequence at the accumulation point of the Feigenbaum cascade. In all of them, the infinite sequence is obtained through a recursive procedure of duplication. The properties of the ruler sequence, in particular, those relating to recursiveness and self-containing, are used to achieve a deeper understanding of these four problems. These new representations of the ruler sequence could inspire new studies in the field of discrete mathematics.

A Review Study of Prime Period Perfect Gaussian Integer Sequences

Prime period sequences can serve as the fundamental tool to construct arbitrary composite period sequences. This is a review study of the prime period perfect Gaussian integer sequence (PGIS). When cyclic group {1,2,…,N−1} can be partitioned into k cosets, where N=kf+1 is an odd prime number, the construction of a degree-(k + 1) PGIS can be derived from either matching the flat magnitude spectrum criterion or making the sequence with ideal periodic autocorrelation function (PACF). This is a systematic approach of prime period N=kf+1 PGIS construction, and is applied to construct PGISs with degrees 1, 2, 3 and 5. However, for degrees larger than 3, matching either the flat magnitude spectrum or achieving the ideal PACF encounters a great challenge of solving a system of nonlinear constraint equations. To deal with this problem, the correlation and convolution operations can be applied upon PGISs of lower degrees to generate new PGISs with a degree of 4 and other higher degrees, e.g., 6, 7, 10, 11, 12, 14, 20 and 21 in this paper. In this convolution-based scheme, both degree and pattern of a PGIS vary and can be indeterminate, which is rather nonsystematic compared with the systematic approach. The combination of systematic and nonsystematic schemes contributes great efficiency for constructing abundant PGISs with various degrees and patterns for the associated applications.

Leveraging past information and machine learning to accelerate land disturbance monitoring

Near real-time (NRT) monitoring of land disturbances holds great importance for delivering emergency aid, mitigating negative social and ecological impacts, and distributing resources for disaster recovery. Many past NRT techniques were built upon examining the overall change magnitude of a spectral anomaly with a predefined threshold, namely the unsupervised approach. However, their lack of fully considering spectral change direction, change date, and pre-disturbance conditions often led to low detection sensitivity and high commission errors, especially when only a few satellite observations were available at the early disturbance stage, eventually resulting in a longer lag to produce a reliable disturbance map. For this study, we developed a novel supervised machine learning approach guided by historical disturbance datasets to accelerate land disturbance monitoring. This new approach consisted of two phases. For the first phase, the supervised approach applied retrospective analysis on historical Harmonized Landsat Sentinel-2 (HLS) datasets from 2015 to 2021, combined with several open disturbance products. The disturbance model was constructed for each condition of consecutive anomaly number, with the aim of enhancing the specificity for delineating early-stage disturbance regions. Then, these stage-based models were applied for an NRT scenario to predict disturbance probabilities with 2022 HLS images incrementally on a weekly basis. To demonstrate the capability of this new approach, we developed an operational NRT system incorporating both the unsupervised and supervised approach. Latency and accuracy were evaluated against 3000 samples that were randomly selected from the five most influential disturbance events of the United States in 2022, based on labels and disturbance dates interpreted from daily PlanetScope images. The evaluation showed that the supervised approach required 15 days (since the start of the disturbance event) to reach the plateau of its F1 curve (where most disturbance pixels are detected with high confidence), seven days earlier with roughly 0.2 F1 score improvement compared to the unsupervised approach (0.733 vs. 0.546 F1 score). Further analysis showed the improvement was mainly due to the substantial decrease in commission errors (17.7% vs 44.4%). The latency component analysis illustrated that the supervised approach only took an average of 4.1 days to yield the first disturbance alert at its fastest daily updating speed, owing to its decreased sensitivity lag. This finding highlighted the importance of using past knowledge and machine learning to reduce detection delays for an NRT monitoring task.

On a general divisor problem related to a certain Dedekind zeta-function over a specific sequence of positive integers

We investigate the average behavior of coefficients of the Dirichlet series of positive integral power of the Dedekind zeta-function $\zeta_{\mathbb{K}_3}(s)$ of a non-normal cubic extension $\mathbb{K}_3$ of $\mathbb{Q}$ over a certain sequence of positive integers. More precisely, we prove an asymptotic formula with an error term for the sum\[ \sum_{{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}+a_{6}^{2}\leq {x}}\atop{(a_{1},a_{2},a_{3},a_{4},a_{5},a_{6})\in\mathbb{Z}^{6}}}a_{k,\mathbb{K}_3} (a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}+a_{6}^{2}),\]where $(\zeta_{\mathbb{K}_3}(s))^{k}:=\sum_{n=1}^{\infty}\frac{a_{k,\mathbb{K}_3}(n)}{n^{s}}$.

On the Lindelöf hypothesis for general sequences

AbstractIn a recent paper, Gonek, Graham, and Lee introduced a notion of the Lindelöf hypothesis (LH) for general sequences that coincides with the usual LH for the Riemann zeta function in the case of the sequence of positive integers. They made two conjectures: that LH should hold for every admissible sequence of positive integers, and that LH should hold for the “generic” admissible sequence of positive real numbers. In this paper, we give counterexamples to the first conjecture, and show that the second conjecture can be either true or false, depending on the meaning of “generic”: we construct probabilistic processes producing sequences satisfying LH with probability 1, and we construct Baire topological spaces of sequences for which the subspace of sequences satisfying LH is meagre. We also extend the main result of Gonek, Graham, and Lee, stating that the Riemann hypothesis is equivalent to LH for the sequence of prime numbers, to the context of Beurling generalized number systems.

A Class of Bounded Iterative Sequences of Integers

In this note, we show that, for any real number τ∈[12,1), any finite set of positive integers K and any integer s1≥2, the sequence of integers s1,s2,s3,… satisfying si+1−si∈K if si is a prime number, and 2≤si+1≤τsi if si is a composite number, is bounded from above. The bound is given in terms of an explicit constant depending on τ,s1 and the maximal element of K only. In particular, if K is a singleton set and for each composite si the integer si+1 in the interval [2,τsi] is chosen by some prescribed rule, e.g., si+1 is the largest prime divisor of si, then the sequence s1,s2,s3,… is periodic. In general, we show that the sequences satisfying the above conditions are all periodic if and only if either K={1} and τ∈[12,34) or K={2} and τ∈[12,59).

A comparison of three evolutionary algorithms for group scheduling in theme parks with multitype facilities.

During the peak tourist season, large theme parks often experience a simultaneous influx of visitors, resulting in prolonged waiting times for popular attractions. This extended waiting significantly reduces tourists' satisfaction and may negatively impact their willingness to revisit the theme park. In Taiwan, schools at all levels often plan graduation trips to theme parks for their students. Students are divided into groups and must enter and exit the theme park at the same time. This article presents a new theme park problem with multitype facilities (TPP-MTF) for student groups. Based on the group's preference for theme park facilities, multitype reserved tickets with popular facilities are designed for groups, so groups do not need to wait for the reserved facilities. Since the waiting time for groups can be reduced, the theme park can also obtain ticket fees in advance and estimate the number of visitors to the theme park, so the theme park and the group can achieve a win-win situation. This article proposes a new decoding approach for a random permutation of integer sequence and embeds it into an immune-based algorithm, genetic algorithm, and particle swarm optimization algorithm to solve the TPP-MTF problem. A theme park in Taiwan was taken as an example and numerical results of the three algorithms were analyzed and compared to verify the effectiveness of the proposed algorithms.

A threshold for the best two-term underapproximation by Egyptian fractions

Let G be the greedy algorithm that, for each θ∈(0,1], produces an infinite sequence of positive integers (an)n=1∞ satisfying ∑n=1∞1/an=θ. For natural numbers p<q, let Υ(p,q) denote the smallest positive integer j such that p divides q+j. Continuing Nathanson’s study of two-term underapproximations, we show that whenever Υ(p,q)⩽3, G gives the (unique) best two-term underapproximation of p/q; i.e., if 1/x1+1/x2<p/q for some x1,x2∈N, then 1/x1+1/x2⩽1/a1+1/a2. However, the same conclusion fails for every Υ(p,q)⩾4. Next, we study stepwise underapproximation by G. Let em=θ−∑n=1m1/an be the mth error term. We compare 1/am to a superior underapproximation of em−1, denoted by N/bm (N∈N⩾2), and characterize when 1/am=N/bm. One characterization is am+1⩾Nam2−am+1. Hence, for rational θ, we only have 1/am=N/bm for finitely many m. However, there are irrational numbers such that 1/am=N/bm for all m. Along the way, various auxiliary results are encountered.

Determinants of Toeplitz–Hessenberg Matrices with Generalized Leonardo Number Entries

Abstract Let un = un (k) denote the generalized Leonardo number defined recursively by un = un− 1 + un− 2 + k for n ≥ 2, where u 0 = u 1 = 1. Terms of the sequence un (1) are referred to simply as Leonardo numbers. In this paper, we find expressions for the determinants of several Toeplitz–Hessenberg matrices having generalized Leonardo number entries. These results are obtained as special cases of more general formulas for the generating function of the corresponding sequence of determinants. Special attention is paid to the cases 1 ≤ k ≤ 7, where several connections are made to entries in the On-Line Encyclopedia of Integer Sequences. By Trudi’s formula, one obtains equivalent multi-sum identities involving sums of products of generalized Leonardo numbers. Finally, in the case k = 1, we also provide combinatorial proofs of the determinant formulas, where we make extensive use of sign-changing involutions on the related structures.

Subdivision algorithms with modular arithmetic

We study the de Casteljau subdivision algorithm for Bezier curves and the Lane-Riesenfeld algorithm for uniform B-spline curves over the integers mod m, where m>2 is an odd integer. We place the integers mod m evenly spaced around a unit circle so that the integer k mod m is located at the position on the unit circle ate2πki/m=cos(2kπ/m)+isin(2kπ/m)↔(cos(2kπ/m),sin(2kπ/m)).Given a sequence of integers (s0,…,sm)mod m, we connect consecutive values sjsj+1 on the unit circle with straight line segments to form a control polygon. We show that if we start these subdivision procedures with the sequence (0,1,…,m)mod m, then the sequences generated by these recursive subdivision algorithms spawn control polygons consisting of the regular m-sided polygon and regular m-pointed stars that repeat with a period equal to the minimal integer k such that 2k=±1modm. Moreover, these control polygons represent the eigenvectors of the associated subdivision matrices corresponding to the eigenvalue 2−1modm. We go on to study the effects of these subdivision procedures on more general initial control polygons, and we show in particular that certain control polygons, including the orbits of regular m-sided polygons and the complete graphs of m-sided polygons, are fixed points of these subdivision procedures.

On sequences of integers with small prime factors

On sequences of integers with small prime factors

Asymptotics of Z-convex polyominoes

The degree of convexity of a convex polyomino P is the smallest integer k such that any two cells of P can be joined by a monotone path inside P with at most k changes of direction. In this paper we show that one can compute in polynomial time the number of polyominoes of area n and degree of convexity at most 2 (the so-called Z-convex polyominoes). The integer sequence that we have computed allows us to conjecture the asymptotic number an of Z-convex polyominoes of area n, ɑn ∼ C·exp(π)√11n/4⁄n3/2.

ConCPDP: A Cross‐Project Defect Prediction Method Integrating Contrastive Pretraining and Category Boundary Adjustment

Software defect prediction (SDP) is a crucial phase preceding the launch of software products. Cross‐project defect prediction (CPDP) is introduced for the anticipation of defects in novel projects lacking defect labels. CPDP can use defect information of mature projects to speed up defect prediction for new projects. So that developers can quickly get the defect information of the new project, so that they can test the software project pertinently. At present, the predominant approaches in CPDP rely on deep learning, and the performance of the ultimate model is notably affected by the quality of the training dataset. However, the dataset of CPDP not only has few samples but also has almost no label information in new projects, which makes the general deep‐learning‐based CPDP model not ideal. In addition, most of the current CPDP models do not fully consider the enrichment of classification boundary samples after cross‐domain, leading to suboptimal predictive capabilities of the model. To overcome these obstacles, we present contrastive learning pretraining for CPDP (ConCPDP), a CPDP method integrating contrastive pretraining and category boundary adjustment. We first perform data augmentation on the source and target domain code files and then extract the enhanced data as an abstract syntax tree (AST). The AST is then transformed into an integer sequence using specific mapping rules, serving as input for the subsequent neural network. A neural network based on bidirectional long short‐term memory (Bi‐LSTM) will receive an integer sequence and output a feature vector. Then, the feature vectors are input into the contrastive module to optimise the feature extraction network. The pretrained feature extractor can be fine‐tuned by the maximum mean discrepancy (MMD) between the feature distribution of the source domain and the target domain and the binary classification loss on the source domain. This paper conducts a large number of experiments on the PROMISE dataset, which is commonly used for CPDP, to validate ConCPDP’s efficacy, achieving superior results in terms of F1 measure, area under curve (AUC), and Matthew’s correlation coefficient (MCC).

On hyper-dual vectors and angles with Pell, Pell-Lucas numbers

&lt;p&gt;In this paper, we introduce two types of hyper-dual numbers with components including Pell and Pell-Lucas numbers. This novel approach facilitates our understanding of hyper-dual numbers and properties of Pell and Pell-Lucas numbers. We also investigate fundamental properties and identities associated with Pell and Pell-Lucas numbers, such as the Binet-like formulas, Vajda-like, Catalan-like, Cassini-like, and d'Ocagne-like identities. Furthermore, we also define hyper-dual vectors by using Pell and Pell-Lucas vectors and discuse hyper-dual angles. This extensionis not only dependent on our understanding of dual numbers, it also highlights the interconnectedness between integer sequences and geometric concepts.&lt;/p&gt;

Modelling the Lengths of Quantitative Motifs in Pseudorandom Integer Sequences

This study explores the behaviour of quantitative motifs in random numerical data, focussing on the property of length. One hundred and twenty sequences of pseudorandom integers were generated and segmented into quantitative motifs using the standard definition. The length–frequency relationship was modelled in each case using the Zipf-Alekseev function. Different combinations of sequence length, integer range, and pseudorandom number generator were trialled. The model-fitting was successful in all cases (𝑅2 1 &gt; 0.9 in 118/120 cases and &gt; 0.8 in 2/120), and the model parameters fell in the same range as those obtained in a previous textual study of motifs. Integer range was the main factor affecting the parameter values, with sequence length affecting the degree of spread. Further comparisons between textual and random data are needed, as well as a theoretical explanation of why motifs in random data demonstrate lawful behaviour. In future textual motif studies, particular attention needs to be paid to possible dependencies on value range and text length.

On the central limit theorem for the elephant random walk with gradually increasing memory and random step size

&lt;abstract&gt;&lt;p&gt;In this paper, we investigate an extended version of the elephant random walk model. Unlike the traditional approach where step sizes remain constant, our model introduces a novel feature: step sizes are generated as a sequence of positive independent and identically distributed random variables, and the step of the walker at time $ n+1 $ depends only on the steps of the walker between times $ 1, ..., m_n $, where $ (m_n)_{n\geqslant 1} $ is a sequence of positive integers growing to infinity as $ n $ goes to infinity. Our main results deal with the validity of the central limit theorem for this new variation of the standard ERW model introduced by Schütz and Trimper in $ 2004 $.&lt;/p&gt;&lt;/abstract&gt;

On a sequence of rational numbers with unusual divisibility by a power of 2

In this note we consider the sequence of rational numbers bn = ∑k=1n2k∕k. We show that the power of 2 in the expansion of bn is unusually large, at least n + 1 − log2(n + 1), and that this bound is best possible. The sequence bn, n = 1,2,3,…, is related to the sequence A0031449 in the On-Line Encyclopedia of Integer Sequences.

Portal-site epinephrine injections improve visualisation in arthroscopic rotator cuff repair

ObjectivesThe aim of this study was to evaluate if portal-site injections of 1:200,000 epinephrine improve intraoperative visualisation in arthroscopic rotator cuff repair. MethodsPatients with partial-thickness supraspinatus tears were selected for the study. They were assigned consecutive numbers and were divided into two groups—control group and intervention group. The surgeries were done by a single surgeon. Every odd-numbered patient was injected. Johnson's visibility classification, surgeon 5-point ordinal Likert scale (LS), and other parameters were recorded. ResultsA total of 221 participants (58.4 ​± ​6.1 years) were selected. Intraoperative visibility was better in the intervention group according to Johnson's classification—satisfactory visibility was achieved in 68 of 110 patients (62%, control group) compared to 89 of 111 patients (80%) (p ​= ​0.003). Surgeon LS was superior in the intervention group, with a notable decrease in worsened visibility cases. The operative time did not alter statistically significantly—82.2 ​± ​14.4 ​min for the control group, compared to 80.9 ​± ​10.8 ​min in the intervention group (p ​= ​0.056). No injection-associated complications were recorded. ConclusionsPortal-site injection of diluted epinephrine solution is safe and improves intraoperative visualisation in arthroscopic rotator cuff repair. This addition does not increase operative time. Level of evidenceLevel 3, case–control study.

MIRROR (LEFT-RECURSIVE) GRAY CODE

Here we consider a version of the Binary Reflected Gray Code (BRGC, or Gray code for short), called the mirror Gray code – in parallel and in comparison to the Gray code. We discuss some sources where the two codes are not distinguished and the reasons why. We present our arguments for treating the two codes as distinct, starting with a definition of the mirror Gray code and showing its main properties. Among the most important of these is the relationship between the two codes – the codewords of the mirror Gray code are “left-right mirror images” of those in the Gray code and vice versa. Other arguments we show are the differences in the algorithms for generating the codewords. We also present another approach that involves representing the codewords by integers (their serial numbers), generating and examining the two codes by using their sequences of serial numbers. For this purpose, we use the bijections that relate the pairs of combinatorial objects: binary vectors in a certain order and the corresponding integer sequences and subsets. In this way, we get different algorithms, sequences, and functions for successor and predecessor, for ranking and unranking at the mirror Gray code. We believe that this article provides at least one more perspective, as well as expands and enriches the knowledge of the Gray code.