Search

Leonetti and Luca ['On the iterates of the shifted Euler's function', Bull. Aust. Math. Soc. , to appear] have shown that the integer sequence $(x_n)_{n\geq 1}$ defined by $x_{n+2}=\phi (x_{n+1})+\phi (x_{n})+k$ , where $x_1,x_2\geq 1$ , $k\geq 0$ and $2 \mid k$ , is bounded by $4^{X^{3^{k+1}}}$ , where $X=(3x_1+5x_2+7k)/2$. We improve this result by showing that the sequence $(x_n)$ is bounded by $2^{2X^2+X-3}$ , where $X=x_1+x_2+2k$. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, the authors give a simple review of closed-form, explicit, and recursive formulas and related results for the nth derivative of the power-exponential function x x , establish two closed-form and explicit formulas for partial Bell polynomials at some specific arguments, and present several new closed-form and explicit formulas for the nth derivative of the power-exponential function x x and for related functions and integer sequences. [ABSTRACT FROM AUTHOR]

Given an n-element point set in the plane, in how many ways can it be peeled off until no point remains? Only one extreme point can be removed at a time. The answer obviously depends on the point set. If the points are in convex position, there are exactly n ! ways, which is the maximum number of ways for n points. But what is the minimum number? [ABSTRACT FROM AUTHOR]

The four subspace problem is a known matrix problem, which is equivalent to determining all the indecomposable representations of a poset consisting of four incomparable points. In this paper, we use solutions of this problem and invariants associated with indecomposable projective modules with some suitable Brauer configuration algebras to categorify the integer sequence encoded in the OEIS as A100705 and some related integer sequences. [ABSTRACT FROM AUTHOR]

By considering the number of all choices of signs + and - such that ± α1 ± α2 ± α3 ... ± αn = 0 and the number of sign -- appeared therein, this paper can give the exact value of ... . In addition, without using the Fourier transformation technique, we can also find the exact value of ... . These two integrals are motivated by the work of Andrican and Bragdasar in 2021, Andria and Tomescu in 2002, and Borwein and Borwein in 2001, respectively. [ABSTRACT FROM AUTHOR]

Venkatachala on the one hand, and Avdispahić & Zejnulahi on the other, both studied integer sequences with an unusual sum property defined in a greedy way, and proved many results about them. However, their proofs were rather lengthy and required numerous cases. In this paper, I provide a different approach, via finite automata, that can prove the same results (and more) in a simple, unified way. Instead of case analysis, we use a decision procedure implemented in the free software Walnut. Using these ideas, we can prove a conjecture of Quet and find connections between Quet's sequence and the "married" functions of Hofstadter. • Uses a new technique, automata theory, to prove properties of some greedily-defined integer sequences. • Finds much simpler proofs of some known results in the literature. • Finds a new connection between Quet's sequence and the Hofstadter married functions. [ABSTRACT FROM AUTHOR]

Given a set of n labeled points in general position in the plane, we remove all of its points one by one. At each step, one point from the convex hull of the remaining set is erased. In how many ways can the process be carried out? The answer obviously depends on the point set. If the points are in convex position, there are exactly n! ways, which is the maximum number of ways for n points. But what is the minimum number? It is shown that this number is (roughly) at least 3n\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$3^n$$\end{document} and at most 12.29n\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$12.29^n$$\end{document}. [ABSTRACT FROM AUTHOR]

In this paper, we consider properties of coefficients of a composition of ordinary generating functions. Using the notion of a composita, we get some new properties for this composition. The obtained properties can be used for distinguishing prime numbers from composite numbers. As an application, the obtained results can be used for getting new primality criteria. For example, we show how to get primality criteria for the Mersenne numbers, the Lucas numbers, the Pell-Lucas numbers, the Jacobsthal-Lucas numbers, and other. [ABSTRACT FROM AUTHOR]

For a positive integer d, a non-negative integer n and a non-negative integer h ≤ n , we study the number C n (d) of principal ideals; and the number C n , h (d) of principal ideals generated by an element of rank h, in the d-tonal partition monoid on n elements. We compute closed forms for the first family, as partial cumulative sums of known sequences. The second gives an infinite family of new integral sequences. We discuss their connections to certain integral lattices as well as to combinatorics of partitions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a class of restricted compositions. We prove that this class appears as a combinatorial interpretation of a generalized third-order recursive sequences. Further identities are also proved. In particular, we show that the difference of the n th r -tribonacci number and the sum of its three predecessors is the n th regular polytopic number. [ABSTRACT FROM AUTHOR]

We consider the number of passes a permutation needs to take through a stack if we only pop the appropriate output values and start over with the remaining entries in their original order. We define a permutation π to be k -pass sortable if π is sortable using k passes through the stack. Permutations that are 1 -pass sortable are simply the stack sortable permutations as defined by Knuth. We define the permutation class of 2 -pass sortable permutations in terms of their basis. We also show all k -pass sortable classes have finite bases by giving bounds on the length of a basis element of the permutation class for any positive integer k. Finally, we define the notion of tier of a permutation π to be the minimum number of passes after the first pass required to sort π. We then give a bijection between the class of permutations of tier t and a collection of integer sequences studied by Parker [The combinatorics of functional composition and inversion, PhD thesis, Brandeis University (1993)]. This gives an exact enumeration of tier t permutations of a given length and thus an exact enumeration for the class of (t + 1) -pass sortable permutations. Finally, we give a new derivation for the generating function in [S. Parker, The combinatorics of functional composition and inversion, PhD thesis, Brandeis University (1993)] and an explicit formula for the coefficients. [ABSTRACT FROM AUTHOR]

We study a recursively defined sequence which is constructed using the least common multiple. Several authors have conjectured that every term of that sequence is or a prime. In this paper we show that this claim is connected to a strong version of Linnik's theorem, which is still unproved. We also study a generalization that replaces the first term by any positive integer. Under this variation some composite numbers may appear now. We give a full characterization of these numbers. [ABSTRACT FROM AUTHOR]

Let Ф(x) = xd + c be an integral polynomial of degree at least 2, and consider the sequence (Фn(0))n=0∞, which is the orbit of 0 under iteration by Ф. Let Dd;c denote the set of positive integers n for which n j Фn(0). We give a characterization of Dd,c in terms of a directed graph and describe a number of its properties, including its cardinality and the primes contained therein. In particular, we study the question of which primes p have the property that the orbit of 0 is a single p-cycle modulo p. We show that the set of such primes is finite when d is even, and conjecture that it is infinite when d is odd. [ABSTRACT FROM AUTHOR]

The Hankel determinants of a given power series f can be evaluated by using the Jacobi continued fraction expansion of f . However the existence of the Jacobi continued fraction needs that all Hankel determinants of f are nonzero. We introduce Hankel continued fraction , whose existence and uniqueness are guaranteed without any condition for the power series f . The Hankel determinants can also be evaluated by using the Hankel continued fraction. It is well known that the continued fraction expansion of a quadratic irrational number is ultimately periodic. We prove a similar result for power series. If a power series f over a finite field satisfies a quadratic equation, then the Hankel continued fraction is ultimately periodic. As an application, we derive the Hankel determinants of several automatic sequences, in particular, the regular paperfolding sequence. Thus we provide an automatic proof of a result obtained by Guo, Wu and Wen, which was conjectured by Coons–Vrbik. [ABSTRACT FROM AUTHOR]

• Proposed generalized cryptographic image processing approaches using integer-series transformation for solar power optimization. • The correlation scatter plots, histogram analysis, and encryption quality metrics are used to assess the performance of the proposed approaches. • The proposed strategies are tested for symmetrical and asymmetrical array sizes under 24 various shading conditions. • The performance of the proposed approaches is compared with the 20 existing methodologies. • The experimental hardware setup of the proposed configurations has been developed for a 4 × 4 PV array and investigated outdoors and in the indoor laboratory under various conditions. Extracting the optimal power from the PV arrays under partial shading conditions (PSC) is challenging. Despite employing bypass diodes, maximum power point trackers, and current injection strategies to mitigate the shading impact, the array yields a considerably sub-optimal output. So, array reconfiguration strategies are employed to overcome this. However, most of the existing static reconfiguration strategies suffer scalability issues, perform arbitrary shade dispersal, and yield distorted array characteristics. Hence, this paper proposes novel PV array configurations based on three distinct image processing strategies employing the integer sequence-based transformation to optimize the array output under PSC. The effectiveness of the proposed strategies is examined using the encryption parameters, histogram analysis and correlation scatter plots. Further, they are implemented for 9 × 9 symmetrical and 5 × 10 unsymmetrical PV arrays and their performance is compared with the 20 existing strategies under 24 distinct shading conditions. Besides, the proposed reconfiguration approaches are validated experimentally for a 4 × 4 PV array in the laboratory and outdoor environments. The qualitative comparison of proposed strategies with state-of-art techniques is discussed in detail. The simulation and hardware results confirm the competency and superiority of the proposed strategies in optimally reconfiguring the PV array yielding the enhancement in output by 32.62%, 20.66%, 16.67%, 14.79%, 13.68%, 13.48% under distinct cases. [ABSTRACT FROM AUTHOR]

We have exhaustively enumerated all simple, connected graphs of a finite order and have computed a selection of invariants over this set. Integer sequences were constructed from these invariants and checked against the Online Encyclopedia of Integer Sequences (OEIS). 141 new sequences were added and six sequences were extended. From the graph database, we were able to programmatically suggest relationships among the invariants. It will be shown that we can readily visualize any sequence of graphs with a given criteria. The code has been released as an open-source framework for further analysis and the database was constructed to be extensible to invariants not considered in this work. [ABSTRACT FROM AUTHOR]

Text The sequence of middle divisors is shown to be unbounded. For a given number n , a n , 0 is the number of divisors of n between n / 2 and 2 n . We explicitly construct a sequence of numbers n ( i ) and a list of divisors in the interesting range, so that the length of the list goes to infinity as i increases. Video For a video summary of this paper, please visit https://youtu.be/drODtRj0gjM . [ABSTRACT FROM AUTHOR]

We find the asymptotic behaviour of a sequence considered by Shaun Cooper. [ABSTRACT FROM AUTHOR]

Define to be the complexity of n, the smallest number of ones needed to write n using an arbitrary combination of addition and multiplication. John Selfridge showed that for all n. Define the defect of n, denoted by , to be ; in this paper we present a method for classifying all nwith for a given r. From this, we derive several consequences. We prove that for with m and k not both zero, and present a method that can, with more computation, potentially prove the same for larger m. Furthermore, defining to be the number of n with and , we prove that , allowing us to conclude that the values of can be arbitrarily large. [ABSTRACT FROM AUTHOR]

For a finite set of points X on the unit hypersphere in ℝd we consider the iteration ui+1 = ui + χi, where χi is the point of X farthest from ui. Restricting to the case where the origin is contained in the convex hull of X we study the maximal length of ui. We give sharp upper bounds for the length of ui independently of X. Precisely, this upper bound is infinity for d ≥ 3 and for d = 2. [ABSTRACT FROM AUTHOR]

The P-positions of the 2-pile take-away game of Wythoff Nim lie on two beams of slope and respectively. We study extensions to this game where a player may also remove simultaneously tokens from either of the piles and from the other, where are given positive integers and where t ranges over the positive integers. We prove that for certain pairs ( p, q) the P-positions are identical to those of Wythoff Nim, but for they do not even lie on two beams. By several experimental results, we conjecture a classification of all pairs ( p, q) for which Wythoff Nim's beams of P-positions transform via a certain splitting behavior, similar to that of going from 2-pile Nim to Wythoff Nim. [ABSTRACT FROM AUTHOR]

Abstract: We prove the following conjecture of Zeilberger. Denoting by the Catalan number, define inductively by and . Then (hence ) is a positive integer. [Copyright &y& Elsevier]

Abstract: We study the smallest possible number of points in a topological space having k open sets. Equivalently, this is the smallest possible number of elements in a poset having k order ideals. Using efficient algorithms for constructing a topology with a prescribed size, we show that this number has a logarithmic upper bound. We deduce that there exists a topology on n points having k open sets, for all k in an interval which is exponentially large in n. The construction algorithms can be modified to produce topologies where the smallest neighborhood of each point has a minimal size, and we give a range of obtainable sizes for such topologies. [Copyright &y& Elsevier]

Abstract: We propose a generic model for the “weighted voting” aggregation step performed by several methods in supervised classification. Further, we construct an algorithm to enumerate the number of distinct aggregate classifiers that arise in this model. When there are only two classes in the classification problem, we show that a class of functions that arises from aggregate classifiers coincides with the class of self-dual positive threshold Boolean functions. [Copyright &y& Elsevier]

Abstract: The technique of determining a generating function for an unambiguous context-free language is known as the Schützenberger methodology. For regular languages, Elena Barcucci et al. proposed an approach for inverting this methodology based on Soittola’s theorem. This idea allows a combinatorial interpretation (by means of a regular language) of certain positive integer sequences that are defined by C-finite recurrences. In this paper we present a Maple implementation of this inverse methodology and describe various applications. We give a short introduction to the underlying theory, i.e., the question of deciding -rationality. In addition, some aspects and problems concerning the implementation are discussed; some examples from combinatorics illustrate its applicability. [Copyright &y& Elsevier]

We introduce the notion of a near-inverse of a non-decreasing sequence of positive integers; near-inverses are intended to assume the role of inverses in cases when the latter cannot exist. We prove that the near-inverse of such a sequence is unique; moreover, the relation of being near-inverses of each other is symmetric, i.e . if sequence g is the near-inverse of sequence f , then f is the near-inverse of g . There is a connection, by approximations, between near-inverses of sequences and inverses of continuous strictly increasing real-valued functions which can be exploited to derive simple expressions for near-inverses. [ABSTRACT FROM AUTHOR]

Abstract: In this study, we compute the spectral norm of the matrix , where are ith components of any integer sequence . Also, we give a corollary and examples related to spectral norm. [Copyright &y& Elsevier]

Abstract: We prove a formal power series identity, relating the arithmetic sum-of-divisors function to commuting triples of permutations. This establishes a conjecture of Franklin T. Adams-Watters. [Copyright &y& Elsevier]

In this brief note, we formulate some modifications of the Pell-Jacobsthal numbers, which belong to a more general class, the Lucas sequences. [ABSTRACT FROM AUTHOR]