Pascal's Triangle Research Articles

Perbandingan Cara Mencacah Banyak Lintasan Terpendek antara Dua Titik pada Grid

The problem of counting the number of shortest paths between two points on a grid is a combinatorial problem that is often found in mathematical competitions or enrichment problems for various levels of school. There are several ways to count, namely by drawing a figure (visual representation), by the number pattern in the Pascal Triangle, and the combination formula. This paper discusses a comparison of each ways related to the problem at hand. On a simple grid all of the three methods can be used. On complex grids the combination formula is the only method can be applied, in some cases the technique of number pattern in Pascal Triangle is still applicable. On the grid with varied obstacles, foresight and flexibility is needed in the selection of the counting methods.

Powers of integers with arithmetic tables

Any powers of $11$ are easily obtained from the Pascal triangle. In this work we study powering and negative powering of any $k$ digit integers by means of certain arithmetic tables.

Triangle method for constructing Zhegalkin polynomial: connection with Pascal's triangle

Introduced by Soviet scientist I. Zhegalkin in 1927, Zhegalkin polynomial is a way to represent a Boolean function as an exclusive or of conjunctions of variables. One of the known algorithms for constructing Zhegalkin polynomial is so called ‘triangle method’ proposed in 1985–1987 by Soviet mathematician V.P. Suprun. Applying of the triangle method mainly coincides with step-by-step constructing Pascal triangle rows using the equality Cn+1k+1 = Cnk + Cnk+1. Therefore, it would be natural to expect for the relation between the calculation of Zhegalkin polynomial by the triangle method and an arrangement of binomial coefficients in Pascal triangle. In this paper, the connection between the triangle method and constructing of Pascal triangle is studied. Besides that, a rather simple proof of the triangle method correctness is proposed. This proof is based on juxtaposition of the triangle method steps and a step-by-step construction of Pascal triangle.

Арифметическая таблица как неотъемлемая часть всей вычислительной математики

The paper is devoted to studying the following issue as a statement. What do we know and what we don’t know about arithmetic tables. Perhaps there is no mathematical problem as naive or simple as finding a method for creating arithmetic tables. We confirm that the general method has not been found yet. This study provides nonterminal solution to this problem. Why? The presentation of arithmetic material in essence, plus some accompanying ideas, makes it possible to develop them further in the system. Materials and methods. The system looks like this: a numerical table as a Pascal's triangle and a symmetric polynomial in two or three variables. Some arithmetic properties of such tables will be found, studied and proved. All this was made possible only after successful decryption of the entire class of numeric tables of truncated triangles in the cryptographic system. Results. For example, the arithmetic properties of truncated Pascal’s triangle for finding all prime numbers have been found and presented, and then their formulas have been placed. In addition to elementary addition and subtraction tables, unlimited “comparison” tables of numbers are given and presented for the first time. Conclusions. For computer implementation of the objectives set, the rules of real actions that should exist for tables have been laid down. Only recurrent numeric series should be used for this purpose.

Irreducible calibrated representations of periplectic Brauer algebras and hook representations of the symmetric group

We construct an infinite tower of irreducible calibrated representations of periplectic Brauer algebras on which the cup-cap generators act by nonzero matrices. As representations of the symmetric group, these are exterior powers of the standard representation (i.e. hook representations). Our approach uses the recently-defined degenerate affine periplectic Brauer algebra, which plays a role similar to that of the degenerate affine Hecke algebra in representation theory of the symmetric group. We write formulas for the representing matrices in the basis of Jucys–Murphy eigenvectors and we completely describe the spectrum of these representations. The tower formed by these representations provides a new, non-semisimple categorification of Pascal's triangle. Along the way, we also prove some basic results about calibrated representations of the degenerate affine periplectic Brauer algebra.

KARAKTERISTIK SEGITIGA LUCAS

Lucas triangle is an array of coeficients of a polynomial forming a pattern which is similar to Pascal triangle. This research studies Lucas triangle and its properties. The research results show that every row in Lucas triangle is begun by the number 1 and is ended by the number 2, the sum of the first n terms of number of 1th column is equal to the number at th row, 2nd column. Besides, the number at nth row and th column of Lucas triangle is for , the sum of the first n terms of number of jth column is equal to the number at th row, column for . The number of Lucas triangle is the sum of two number terms in preceded row, that is the number at th row, and the number at th row, . Then, the sum of coefficients of each row of Lucas triangle is .
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Maximal Spectral Efficiency of OFDM With Index Modulation Under Polynomial Space Complexity

In this letter, we demonstrate a mapper that enables all waveforms of OFDM with Index Modulation (OFDM-IM) while preserving polynomial time and space computational complexities. Enabling all OFDM-IM waveforms maximizes the spectral efficiency (SE) gain over the classic OFDM but, as far as we know, the computational overhead of the resulting mapper remains conjectured as prohibitive across the OFDM-IM literature. We show that the largest number of binomial coefficient calculations performed by the original OFDM-IM mapper is polynomial on the number of subcarriers, even under the setup that maximizes the SE gain over OFDM. Also, such coefficients match the entries of the so-called Pascal's triangle (PT). Thus, by assisting the OFDM-IM mapper with a PT table, we show that the maximum SE gain over OFDM can be achieved under polynomial (rather than exponential) time and space complexities.

A Pascal Triangle Type Calculation for a Particular Infinite Series

In general, finding the sum of an infinite series is not always possible. However, there are infinite series whose sums can be computed. In our earlier paper, we derived a formula for computing the sum of a type of polylogarithm series involving multinomial coefficients. In this paper, we show that the formula leads to an elementary computation for the series $\sum_{k=1}^{\infty}\frac{k^n}{a^k}$ involving numbers obtained by a method similar to Pascal's triangle. We also show that our result is the number of ways of distributing $n$ distinct objects in $n$ or fewer distinct nonempty cells.

Assembling Shape-Persistent High-Order Sierpiński Triangular Fractals.

SummaryFractals are a series of intricate patterns with aesthetic, mathematic, and philosophic significance. The Sierpiński triangles have been known for more than one hundred years, but only recently discrete shape-persistent low-generation (mainly ST-1) fractal supramolecules have been realized. Herein, we report a retro-assembly pathway to the nanometer-scale, supra-macromolecular second-generation Sierpiński triangle and its third-generation saturated counterpart (Pascal's triangle). These gigantic triangular assemblies are unambiguously confirmed by NMR, DOSY, ESI-MS, TWIM-MS, TEM, and AFM analyses. Notably, the dense-packed counterions of these discrete triangular architectures could further form supramolecular hydro-gels in water. This work not only provides a fundamental chemical pathway to explore various giant supramolecular constructs and further overcome the synthetic limitation of complicated molecular fractals, but also presents a new type of supramolecular hydro-gels with potential in controlled release applications.

Fabrication of Pascal-triangle Lattice of Proteins by Inducing Ligand Strategy.

A protein Pascal triangle has been constructed as new type of supramolecular architecture by using the inducing ligand strategy that we previously developed for protein assemblies. Although mathematical studies on this famous geometry have a long history, no work on such Pascal triangles fabricated from native proteins has been reported so far due to their structural complexity. In this work, by carefully tuning the specific interactions between the native protein building block WGA and the inducing ligand R‐SL, a 2D Pascal‐triangle lattice with three types of triangular voids has been assembled. Moreover, a 3D crystal structure was obtained based on the 2D Pascal triangles. The distinctive carbohydrate binding sites of WGA and the intralayer as well as interlayer dimerization of RhB was the key to facilitate nanofabrication in solution. This strategy may be applied to prepare and explore various sophisticated assemblies based on native proteins.

Fabrication of Pascal‐triangle Lattice of Proteins by Inducing Ligand Strategy

AbstractA protein Pascal triangle has been constructed as new type of supramolecular architecture by using the inducing ligand strategy that we previously developed for protein assemblies. Although mathematical studies on this famous geometry have a long history, no work on such Pascal triangles fabricated from native proteins has been reported so far due to their structural complexity. In this work, by carefully tuning the specific interactions between the native protein building block WGA and the inducing ligand R‐SL, a 2D Pascal‐triangle lattice with three types of triangular voids has been assembled. Moreover, a 3D crystal structure was obtained based on the 2D Pascal triangles. The distinctive carbohydrate binding sites of WGA and the intralayer as well as interlayer dimerization of RhB was the key to facilitate nanofabrication in solution. This strategy may be applied to prepare and explore various sophisticated assemblies based on native proteins.

Fractal Generalized Pascal Matrices

The set of generalized Pascal matrices whose entries are generalized binomial coefficients is regarded as a group with respect to Hadamard multiplication. A special system of matrices is introduced and is used to construct fractal generalized Pascal matrices. The Pascal matrix (triangle) is expanded in theHadamard product of fractal generalized Pascal matrices whose nonzero entries are pk, where p is a fixed prime and k = 0, 1, 2,.... The introduced system of matrices suggests the idea of “zero” generalized Pascal matrices, each of which is the limit case of a certain set of generalized Pascal matrices. “Zero” fractal generalized Pascal matrices, which are exemplified by the Pascal triangle modulo 2, are considered.

Divisibility Patterns within Pascal Divisibility Networks

The Pascal triangle is so simple and rich that it has always attracted the interest of professional and amateur mathematicians. Their coefficients satisfy a myriad of properties. Inspired by the work of Shekatkar et al., we study the divisibility patterns within the elements of the Pascal triangle, through its decomposition into Pascal’s matrices, from the perspective of network science. Applying Kolmogorov–Smirnov test, we determine that the degree distribution of the resulting network follows a power-law distribution. We also study degrees, global and local clustering coefficients, stretching graph, averaged path length and the mixing assortative.

An energetic boundary functional method for solving the inverse heat conductivity problems in arbitrary plane domains

In the paper, we solve Caldero´n’s inverse conductivity problems in a rectangular plane domain and in an arbitrary plane domain, to recover an unknown heat conductivity function σ(x, y) inside a material object. Firstly, the homogenization functions are derived, by which the Dirichlet as well as the Neumann boundary conditions over-specified on the boundary become zeros for the new variable. Secondly, an integral identity expressing the conservation of energy is derived; hence, we can set up an energy functional to effectively solve the inverse heat conductivity problem. Thirdly, a sequence of boundary functions in terms of the modified Pascal triangle are given, which satisfy the homogeneous boundary conditions automatically. A new methodology of energetic boundary functional is constructed in a linear space, wherein each energetic boundary function preserves the energy identity. The linear system used to recover the unknown heat conductivity function σ(x, y) with energetic boundary functions as the bases is derived, and then the iterative algorithm is developed which converges very fast. The accuracy and robustness of the energetic boundary functional method (EBFM) are confirmed by comparing the recovered results under a large noisy disturbance to the exact heat conductivity functions σ(x, y).

The Pseudo-Pascal Triangle of Maximum Deng Entropy

PPascal triangle (known as Yang Hui Triangle in Chinese) is an important model in mathematics while the entropy has been heavily studied in physics or as uncertainty measure in information science. How to construct the the connection between Pascal triangle and uncertainty measure is an interesting topic. One of the most used entropy, Tasllis entropy, has been modelled with Pascal triangle. But the relationship of the other entropy functions with Pascal triangle is still an open issue. Dempster-Shafer evidence theory takes the advantage to deal with uncertainty than probability theory since the probability distribution is generalized as basic probability assignment, which is more efficient to model and handle uncertain information. Given a basic probability assignment, its corresponding uncertainty measure can be determined by Deng entropy, which is the generalization of Shannon entropy. In this paper, a Pseudo-Pascal triangle based the maximum Deng entropy is constructed. Similar to the Pascal triangle modelling of Tasllis entropy, this work provides the a possible way of Deng entropy in physics and information theory.

Infimal Convolution and Duality in Problems with Third-Order Discrete and Differential Inclusions

This paper is concerned with the Mayer problem for third-order evolution differential inclusions; to this end, first we use auxiliary problems with third-order discrete and discrete-approximate inclusions. In the form of Euler–Lagrange-type inclusions and transversality conditions, necessary and sufficient optimality conditions are derived. The principal idea of obtaining optimal conditions is locally adjoint mappings. Then, applying infimal convolution concept of convex functions, step by step, we construct the dual problems for third-order discrete, discrete-approximate and differential inclusions and prove duality results. It appears that the Euler–Lagrange-type inclusions are duality relations for both primary and dual problems and that the dual problem for discrete-approximate problem makes a bridge between the dual problems of discrete and continuous problems. As a result, the passage to the limit in the dual problem with discrete approximations plays a substantial role in the next investigations, without which it is hardly ever possible to establish any duality to continuous problem. In this way, relying to the described method for computation of the conjugate and support functions of discrete-approximate problems, a Pascal triangle with binomial coefficients can be successfully used for any “higher order” calculations. Thus, to demonstrate this approach, some semilinear problems with discrete and differential inclusions of third order are considered. These problems show that maximization in the dual problems is realized over the set of solutions of the Euler–Lagrange-type discrete/differential inclusions.

-DEFORMED RATIONALS AND -CONTINUED FRACTIONS

We introduce a notion of$q$-deformed rational numbers and$q$-deformed continued fractions. A$q$-deformed rational is encoded by a triangulation of a polygon and can be computed recursively. The recursive formula is analogous to the$q$-deformed Pascal identity for the Gaussian binomial coefficients, but the Pascal triangle is replaced by the Farey graph. The coefficients of the polynomials defining the$q$-rational count quiver subrepresentations of the maximal indecomposable representation of the graph dual to the triangulation. Several other properties, such as total positivity properties,$q$-deformation of the Farey graph, matrix presentations and$q$-continuants are given, as well as a relation to the Jones polynomial of rational knots.

Non Degeneration of Fibonacci Series, Pascal’s Elements and Hex Series

Generally Fibonacci series and Lucas series are the same, they converge to golden ratio. After I read Fibonacci series, I thought, is there or are there any series which converges to golden ratio. Because of that I explored the inter relations of Fibonacci series when I was intent on Fibonacci series in my difference parallelogram. In which, I found there is no degeneration on Fibonacci series. In my thought, Pascal triangle seemed like a lower triangular matrix, so I tried to find the inverse for that. In inverse form, there is no change against original form of Pascal elements matrix. One day I played with ring magnets, which forms hexagonal shapes. Number of rings which forms Hexagonal shape gives Hex series. In this paper, I give the general formula for generating various types of Fibonacci series and its non-degeneration, how Pascal elements maintain its identities and which shapes formed by hex numbers by difference and matrices.

PENERAPAN METODE SEGITIGA PASCAL ALJABAR UNTUK MEMAKSIMALKAN TEKNIK PENYANDIAN ALGORITMA WORD AUTO KEY ENCRYPTION (WAKE)

Security of data stored on computers is now an absolute requirement, because every data has a high enough value for the user, reader and owner of the data itself. To prevent misuse of the data by other parties, data security is needed. Data security is the protection of data in a system against unauthorized authorization, modification, or destruction. The science that explains the ways of securing data is known as cryptography, while the steps in cryptography are called critical algorithms. At this time, there are many cryptographic algorithms whose keys are weak especially the symmetric key algorithm because they only have one key, the key for encryption is the same as the decryption key so it needs to be modified so that the cryptanalysts are confused in accessing important data. The cryptographic method of Word Auto Key Encryption (WAKE) is one method that has been used to secure data where in this case the writer wants to maximize the encryption key and description of the WAKE algorithm that has been processed through key formation. One way is to apply the algebraic pascal triangle method to maximize the encryption key and description of the WAKE algorithm, utilizing the numbers contained in the columns and rows of the pascal triangle to make shifts on the encryption key and the description of the WAKE algorithm.Keywords: Cryptography, WAKE, pascal

Modified For The Pascal Triangle Multinomial

Newton’s Binomial Theorem applied at the rate of 2 with the formula: (a1+a2)n=∑r=0nC(n,r)a1n−ra2r Problems in algebra are not limited binomial. Binomial only is not enough, so that multinomial is necessary. Multinomial Theorem has the formula: (a1+a2+…+ak)n=∑n1,n2,…,nk≥0n!n1!n2!…nk!a1n1a2n2…aknk The use of theorem in binomial problem is less practical so that Pascal Triangle is prefered, for easier use Pascal’s Triangle. Solution of Triangle with the theorem multinomial problem more complicated. By analyzing multinomial through binomial form, can be obtained from modification that allows the Pascal triangle. The focus of the discussion is to determine Pascal Modified Triangular shape of a multinomial. This basic research using descriptive method, by analyzing the relevant theory is based on the study of literature. The results obtained are Modified Pascal’s Triangle, which facilitates the work in the multinomial.