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Journal articles on the topic 'Binomial Triangle'

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Published: 2 June 2025

Last updated: 31 July 2025

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1

Lewis, Barry. "Generalising Pascal's Triangle." Mathematical Gazette 88, no. 513 (2004): 447–56. http://dx.doi.org/10.1017/s0025557200176089.

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Pascal's triangle, the Binomial expansion and the recurrence relation between its entries are all inextricably linked. In the normal course of events, the Binomial expansion leads to Pascal's Triangle, and thence to the recurrence relation between its entries. In this article we are going to reverse this process to make it possible to explore a particular type of generalisation of such interlinked structures, by generalising the recurrence relation and then exploring the resulting generalised ‘Pascal Triangle’ and ‘Binomial expansion’. Within the spectrum of generalisations considered, we find

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2

Barnes, Benedict, E. D. J. O. Wusu-Ansah, S. K. Amponsah, and I. A. Adjei. "The Proofs of Triangle Inequality Using Binomial Inequalities." European Journal of Pure and Applied Mathematics 11, no. 1 (2018): 352–61. http://dx.doi.org/10.29020/nybg.ejpam.v11i1.3165.

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In this paper, we introduce the different ways of proving the triangle inequality ku − vk ≤ kuk + kvk, in the Hilbert space. Thus, we prove this triangle inequality through the binomial inequality and also, prove it through the Euclidean norm. The first generalized procedure for proving the triangle inequality is feasible for any even positive integer n. The second alternative proof of the triangle inequality establishes the Euclidean norm of any two vectors in the Hilbert space.

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3

M, Mahalakshmi, Kannan J, Deepshika A, and Kaleeswari K. "Existence and Non - Existence of Exponential Diophantine Triangles Over Triangular Numbers." Indian Journal of Science and Technology 16, no. 41 (2023): 3599–604. https://doi.org/10.17485/IJST/v16i41.2338.

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Abstract <strong>Objectives:</strong> The specified problem addressed here is the existence and non-existence of Exponential Diophantine triangles over triangular numbers ( ). <strong>Methods:</strong> An Exponential Diophantine triangle over triangular numbers is defined as a triangle with sides and where and are non - negative integers such that . To prove the existence of such triangles, negative Pell’s equation and its solutions are used along with some basic number theoretic concepts. To verify the non-existence, the well-known Catalan’s conjecture, binomial exp

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4

M, Mahalakshmi, Kannan J, Deepshika A, and Kaleeswari K. "Existence and Non - Existence of Exponential Diophantine Triangles Over Triangular Numbers." Indian Journal of Science and Technology 16, no. 41 (2023): 3599–604. https://doi.org/10.17485/IJST/v16i41.2338.

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Abstract <strong>Objectives:</strong> The specified problem addressed here is the existence and non-existence of Exponential Diophantine triangles over triangular numbers ( ). <strong>Methods:</strong> An Exponential Diophantine triangle over triangular numbers is defined as a triangle with sides and where and are non - negative integers such that . To prove the existence of such triangles, negative Pell’s equation and its solutions are used along with some basic number theoretic concepts. To verify the non-existence, the well-known Catalan’s conjecture, binomial exp

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M, Mahalakshmi, Kannan J, Deepshika A, and Kaleeswari K. "Existence and Non - Existence of Exponential Diophantine Triangles Over Triangular Numbers." Indian Journal of Science and Technology 16, no. 41 (2023): 3599–604. https://doi.org/10.17485/IJST/v16i41.2338.

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Abstract <strong>Objectives:</strong> The specified problem addressed here is the existence and non-existence of Exponential Diophantine triangles over triangular numbers ( ). <strong>Methods:</strong> An Exponential Diophantine triangle over triangular numbers is defined as a triangle with sides and where and are non - negative integers such that . To prove the existence of such triangles, negative Pell’s equation and its solutions are used along with some basic number theoretic concepts. To verify the non-existence, the well-known Catalan’s conjecture, binomial exp

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6

M, Mahalakshmi, Kannan J, Deepshika A, and Kaleeswari K. "Existence and Non - Existence of Exponential Diophantine Triangles Over Triangular Numbers." Indian Journal of Science and Technology 16, no. 41 (2023): 3599–604. https://doi.org/10.17485/IJST/v16i41.2338.

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Abstract <strong>Objectives:</strong> The specified problem addressed here is the existence and non-existence of Exponential Diophantine triangles over triangular numbers ( ). <strong>Methods:</strong> An Exponential Diophantine triangle over triangular numbers is defined as a triangle with sides and where and are non - negative integers such that . To prove the existence of such triangles, negative Pell’s equation and its solutions are used along with some basic number theoretic concepts. To verify the non-existence, the well-known Catalan’s conjecture, binomial exp

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7

M, Mahalakshmi, Kannan J, Deepshika A, and Kaleeswari K. "Existence and Non - Existence of Exponential Diophantine Triangles Over Triangular Numbers." Indian Journal of Science and Technology 16, no. 41 (2023): 3599–604. https://doi.org/10.17485/IJST/v16i41.2338.

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Abstract <strong>Objectives:</strong> The specified problem addressed here is the existence and non-existence of Exponential Diophantine triangles over triangular numbers ( ). <strong>Methods:</strong> An Exponential Diophantine triangle over triangular numbers is defined as a triangle with sides and where and are non - negative integers such that . To prove the existence of such triangles, negative Pell’s equation and its solutions are used along with some basic number theoretic concepts. To verify the non-existence, the well-known Catalan’s conjecture, binomial exp

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8

Ahmia, Moussa, and Hacène Belbachir. "Preserving log-concavity for p,q-binomial coefficient." Discrete Mathematics, Algorithms and Applications 11, no. 02 (2019): 1950017. http://dx.doi.org/10.1142/s1793830919500174.

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We study the log-concavity of a sequence of [Formula: see text]-binomial coefficients located on a ray of the [Formula: see text]-Pascal triangle for certain directions, and we establish the preserving log-concavity of linear transformations associated to [Formula: see text]-Pascal triangle.

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9

Zhu, Bao-Xuan. "Log-concavity and strong q-log-convexity for Riordan arrays and recursive matrices." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 147, no. 6 (2017): 1297–310. http://dx.doi.org/10.1017/s0308210516000500.

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Let [An,k]n,k⩾0 be an infinite lower triangular array satisfying the recurrencefor n ⩾ 1 and k ⩾ 0, where A0,0 = 1, A0,k = Ak,–1 = 0 for k > 0. We present some criteria for the log-concavity of rows and strong q-log-convexity of generating functions of rows. Our results can be applied to many well-known triangular arrays, such as the Pascal triangle, the Stirling triangle of the second kind, the Bell triangle, the large Schröder triangle, the Motzkin triangle, and the Catalan triangles of Aigner and Shapiro, in a unified approach. In addition, we prove that the binomial transformation not o

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10

Beiu, Valeriu, Leonard Dăuş, Marilena Jianu, Adela Mihai, and Ion Mihai. "On a Surface Associated with Pascal’s Triangle." Symmetry 14, no. 2 (2022): 411. http://dx.doi.org/10.3390/sym14020411.

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An open problem in reliability theory is that of finding all the coefficients of the reliability polynomial associated with particular networks. Because reliability polynomials can be expressed in Bernstein form (hence linked to binomial coefficients), it is clear that an extension of the classical discrete Pascal’s triangle (comprising all the binomial coefficients) to a continuous version (exhibiting infinitely many values in between the binomial coefficients) might be geometrically helpful and revealing. That is why we have decided to investigate the geometric properties of a continuous ext

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11

Guo, Victor J. W., and Jiang Zeng. "Factors of binomial sums from the Catalan triangle." Journal of Number Theory 130, no. 1 (2010): 172–86. http://dx.doi.org/10.1016/j.jnt.2009.07.005.

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12

Fjelstad, P. "The binomial theorem and the extended Pascal's triangle." Computers & Mathematics with Applications 21, no. 9 (1991): 11–17. http://dx.doi.org/10.1016/0898-1221(91)90121-j.

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13

Leroy, Julien, Michel Rigo, and Manon Stipulanti. "Generalized Pascal triangle for binomial coefficients of words." Advances in Applied Mathematics 80 (September 2016): 24–47. http://dx.doi.org/10.1016/j.aam.2016.04.006.

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14

Matsui, H., D. Minematsu, T. Yamauchi, and R. Miy Adera. "Pascal-like triangles and Fibonacci-like sequences." Mathematical Gazette 94, no. 529 (2010): 27–41. http://dx.doi.org/10.1017/s0025557200007129.

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In [1] and [2] we demonstrated how Pascal-like triangles arose from the probabilities associated with the various outcomes of a particular game (see Definition 1 below). It was also shown that they could be considered as generalisations of Pascal's triangle. In this article we show how Fibonacci-like sequences arise from our Pascal-like triangles, and demonstrate the existence of simple relationships between these Fibonacci-like sequences and the Fibonacci sequence itself. In addition we will investigate a generalisation of the binomial coefficients that appears when considering an extended ve

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15

Xuan, Wei, Huawei Cao, Mingyu Yan, Zhimin Tang, Xiaochun Ye, and Dongrui Fan. "BSR-TC: Adaptively Sampling for Accurate Triangle Counting over Evolving Graph Streams." International Journal of Software Engineering and Knowledge Engineering 31, no. 11n12 (2021): 1561–81. http://dx.doi.org/10.1142/s021819402140012x.

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Triangle counting is a fundamental graph mining problem, widely employed in various real-world application scenarios. Given the large scale of graph streams and limited memory space, it is feasible to achieve the estimation of global and local triangles by sampling. Existing streaming algorithms for triangle counting can be generalized into two categories: Reservoir-based methods and Bernoulli-based methods. The former use a fixed memory budget, whose size is difficult to set for accurate estimation without any prior knowledge about graph streams. The latter sample edges by a specified probabi

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16

Aliu, Arta, Shpetim Rexhepi, and Egzona Iseni. "Efficiency of understanding some mathematical problems by means of Pascal’s triangle." International Electronic Journal of Mathematics Education 18, no. 4 (2023): em0753. http://dx.doi.org/10.29333/iejme/13713.

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Various features have been found hidden in the Pascal triangle. In this paper, some very well-known properties of the Pascal triangle will be presented, as well as the properties related to different extensions of the triangle, namely the Pascal pyramid. Given that in the textbooks of the tenth grade, respectively in the school, where we realised the research but also in general in other schools, the importance of the Pascal triangle is not at the right level, then in this paper it has been shown very well that many different exercises in mathematics. The purpose of this paper is to look at th

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Yamagami, Atsushi, and Kazuki Taniguchi. "On a Generalization of a Lucas’ Result and an Application to the 4-Pascal’s Triangle." Symmetry 12, no. 2 (2020): 288. http://dx.doi.org/10.3390/sym12020288.

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The Pascal’s triangle is generalized to “the k-Pascal’s triangle” with any integer k ≥ 2 . Let p be any prime number. In this article, we prove that for any positive integers n and e, the n-th row in the p e -Pascal’s triangle consists of integers which are congruent to 1 modulo p if and only if n is of the form p e m − 1 p e − 1 with some integer m ≥ 1 . This is a generalization of a Lucas’ result asserting that the n-th row in the (2-)Pascal’s triangle consists of odd integers if and only if n is a Mersenne number. As an application, we then see that there exists no row in the 4-Pascal’s tri

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18

Gevorkyan, Yuriy. "Geometric approach to the proof of Fermat's last theorem." EUREKA: Physics and Engineering, no. 4 (July 30, 2022): 127–36. https://doi.org/10.21303/2461-4262.2022.002488.

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A geometric approach to the proof of Fermat's last theorem is proposed. Instead of integers a, b, c, Fermat's last theorem considers a triangle with side lengths a, b, c. It is proved that in the case of right-angled and obtuse-angled triangles Fermat's equation has no solutions. When considering the case when a, b, c are sides of an acute triangle, it is proved that Fermat's equation has no entire solutions for p>2. The numbers a=k, b=k+m, c=k+n, where k, m, n are natural numbers satisfying the inequalities n>m, n

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19

Haeseler, Fritz V., Heinz-Otto Peitgen, and Gencho Skordev. "Pascal's triangle, dynamical systems and attractors." Ergodic Theory and Dynamical Systems 12, no. 3 (1992): 479–86. http://dx.doi.org/10.1017/s0143385700006908.

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AbstractThis paper establishes a global dynamical systems approach for the fractal patterns which are obtained when analysing the divisibility of binomial coefficients modulo a prime power. The general framework is within the class of hierarchical iterated function systems. As a consequence we obtain a complete deciphering of the hierarchical self-similarity features.

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20

Handa, Nidhi, and Padmavati Taneja. "History of Binomial and Multinomial Expansions." International Transactions in Mathematical Sciences and Computer 15, no. 01 (2022): 01–08. http://dx.doi.org/10.58517/itmsc.2022.15101.

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In applied mathematics Binomial Expansion and Multinomial expansion are of great importance. In around 300 BCE Indian mathematician Pingala had derived the method of obtainng a triangular arrangement known as “Meru-Prastar” for attainment of coefficients of binomial expansion. In sixteenth century, CE it was rediscovered by French mathematician Blasé Pascal (1588-1688CE) and termed as Pascal’s triangle. This paper discusses the development of binomial expansion, multinomial expansion with its applications. The paper also emphasizes the fact that the historical roots of binomial expansion are e

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21

Hilton, Peter, and Jean Pedersen. "Stop-sign theorems and binomial coefficients." Mathematical Gazette 94, no. 530 (2010): 247–61. http://dx.doi.org/10.1017/s0025557200006513.

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Dedicated to the memory of Russell Towle, a remarkable man who contributed so much to geometry and to other aspects of the quality of life.We introduce an expanded notation where r + s = n, for the binomial coefficient , and then use this expanded notation to develop theorems involving 8 binomial coefficients, analogous to the Star of David Theorem, which. in its original form, involved the 6 neighbours of a given binomial coefficient in the Pascal Triangle (see Section 3), that appeared in [1,2,3,4,5,6,7,8,9].

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22

Hansen, Karen M., and Geoffrey J. Lewis. "Sharing Teaching Ideas: Finding a Parking Spot for the Binomial Theorem." Mathematics Teacher 101, no. 1 (2007): 46–49. http://dx.doi.org/10.5951/mt.101.1.0046.

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Oftentimes when learning the binomial theorem, students can apply the theorem's formula, but it has little meaning for them. Introducing the binomial theorem using an engaging application gives the theorem life and meaning. This activity relates the coefficients in the formula to finding a parking spot. First, the students discover Pascal's triangle using a parking spot simulation.

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23

Kosobutskyy, Petro, Anastasiia Yedyharova та Taras Slobodzyan. "FROM NEWTON'S BINOMIAL AND PASCAL’S TRIANGLE TO СOLLATZ'S PROBLEM". Computer Design Systems. Theory and Practice 5, № 1 (2023): 121–27. http://dx.doi.org/10.23939/cds2023.01.121.

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It is shown that: 1. The sequence {20,21, 22, 23, 24, 25, 26, 27,28,...} that forms the main graph m=1 of Collatz is related to the power transformation of Newton's binomial (1+1)ξ, ξ=0,1,2,3,... 2. The main Kmain and side m >1 graphs and their corresponding sequences {Kmain} and {Km} are related by the relation {Km}=m⋅{Kmain}. 3. Side graphs generated by prime odd numbers 5, 7, 11, 13, 17, 19, 23, 25, 29, 31,… are not divisible by three, are formed without nodes. Side graphs, which are generated by compozite of odd numbers 3, 9, 15, 21, 27, 33, 39, 45,… are divisible by three, are formed w

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24

Alova, Chard Aye. "On the Proof of Pascal’s Identity using Vedic Mathematics." La Sallianum Ars et Scientia 1, no. 1 (2020): 118–42. http://dx.doi.org/10.70228/cbj2022029.

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This paper presents the proof of Pascal's Identity using the sutras and methods of Vedic Mathematics. From the construction of the proof and the results of the theorem, corollaries have been set up and fundamental properties have been explained. All these finally exhibit its fascinating product-the Pascal’s Triangle. The fundamental properties of the Pascal’s Triangle are also elaborated and discussed using the lens of Vedic Mathematics, such as, binomial coefficients of the binomial theorem, and the property called unimodality. This paper is a result of the establishment of the different form

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25

Cardell, Sara D., and Amparo Fúster-Sabater. "Binomial Representation of Cryptographic Binary Sequences and Its Relation to Cellular Automata." Complexity 2019 (March 24, 2019): 1–13. http://dx.doi.org/10.1155/2019/2108014.

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The binomial sequences are binary sequences that correspond to the diagonals of the binary Sierpinski’s triangle. They have fancy properties such that all the sequences with period equal to a power of 2 can be represented as the sum of a finite set of binomial sequences. Other structural properties of these sequences (period, linear complexity, construction rules, or relations among the different binomial sequences) have been analyzed in detail. Furthermore, this work enhances the close relation between the binomial sequences and a kind of Boolean networks, known as linear cellular automata. I

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Kılıç, Emrah, and Helmut Prodinger. "Identities with squares of binomial coefficients: An elementary and explicit approach." Publications de l'Institut Math?matique (Belgrade) 99, no. 113 (2016): 243–48. http://dx.doi.org/10.2298/pim1613243k.

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In 2014, Slavik presented a recursive method to find closed forms for two kinds of sums involving squares of binomial coefficients. We give an elementary and explicit approach to compute these two kinds of sums. It is based on a triangle of numbers which is akin to the Stirling subset numbers.

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27

عبد الرحمن, محمد السماني. "Generalization of the Fibonacci sequence, Pascal's triangle, and the binomial theorem." المجلة العربية للعلوم و نشر الأبحاث 10, no. 1 (2024): 41–52. http://dx.doi.org/10.26389/ajsrp.n191022.

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هدف هذا البحث إلى تعميم العلاقة بين مثلث باسكال وتوسيع ثنائي باستخدام المتغيرات بدلاً من الأرقام. يتم تشكيل المثلث باستخدام المتغير (d) بدلاً من المصطلح الصفري (0)، والمتغير (a) بدلاً من المصطلح الأول (1)، والمتغير (m) كتعميم لنظرية الثنائي. يتم دراسة الأنماط الرياضية الناتجة عن تشكيل المثلث باستخدام هذه المتغيرات، مما يؤدي إلى خمسة معادلات رياضية جديدة: معادلة عمودية، معادلة الوتر، معادلة الصف، مجموع معادلات الصفوف، ومعادلة تسلسلات الوظائف الذهبية. تُعتبر معادلة تسلسلات الوظائف الذهبية تعميمًا غير مسبوق للمصطلح الثاني لتسلسل فيبوناتشي وتسلسل لوكاس. بالإضافة إلى ذلك، يُصاغ معادلة جديدة وغير مس

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28

Włoch, Iwona, and Andrzej Włoch. "On Some Multinomial Sums Related to the Fibonacci Type Numbers." Tatra Mountains Mathematical Publications 77, no. 1 (2020): 99–108. http://dx.doi.org/10.2478/tmmp-2020-0035.

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Abstract In this paper we investigate Fibonacci type sequences defined by kth order linear recurrence. Based on their companion matrix and its graph interpretation we determine multinomial and binomial formulas for these sequences. Moreover we present a graphical rule for calculating the words of these sequences from the Pascal’s triangle.

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29

Chen, Siqi. "Properties and Applications of Pascal's Triangle and Pascal's Pyramid." Theoretical and Natural Science 84, no. 1 (2025): 36–39. https://doi.org/10.54254/2753-8818/2025.21225.

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With the increasing application of mathematics in modern society, a deep understanding of mathematical foundational theories becomes particularly important. This paper primarily investigates Pascal's Triangle and its three-dimensional extension, Pascal's Pyramid, and explores their properties and applications. The purpose of the study is to show their applications in different math-related areas, including algebra, probability theory, and combinatorial mathematics through literature review and mathematical modeling. This paper used case and data analysis to explore the relationships between Pa

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30

Ibrahim, Ladan Umaru, Aliyu Nuhu Shuaibu, Tanko Ishaya, Ahmed Tijjani Salawudeen, and Kabiru Ahmed. "Location of Points on Plane and the Order of Disposition of Sum of Powers of Cardinal Coordinates." KIU Journal of Science, Engineering and Technology 1, no. 2 (2022): 1–8. http://dx.doi.org/10.59568/kjset-2022-1-2-01.

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This article exposed the structure of all orders of the sum of powers of cardinal coordinates with the explicitly determined computational disposition of similarity, between the coefficient of binomial expansion and the sum of powers of pairs of cardinal points on the plane. The proofs were achieved by a logical deployment of combinatorial techniques, laws of indices on powers of cardinal bases, and a comparison of corresponding results. The results proved conclusively that the sum of the powers of cardinal points is equal to the coefficients of the Binomial expansion with respect to the Pasca

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31

Khassawneh, Bashar, and Benedek Nagy. "Polynomial and Multinomial Coefficients in Terms of Number of Shortest Paths." Proceedings of the Bulgarian Academy of Sciences 75, no. 4 (2022): 495–503. http://dx.doi.org/10.7546/crabs.2022.04.03.

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The binomial coefficients show, in fact, the number of shortest paths in the square grid if only grid paths, i.e., paths on the grid lines, are allowed, and they also give the number of shortest paths in the hexagonal grid. When diagonal steps are also allowed in the square grid, the number of shortest paths can be described by trinomial coefficients. They form a triangle where three neighbour elements in the previous row are summed. We consider also further generalisations of such triangles and their elements, quadrinomial and n-nomial coefficients. In this context, n-nomial coefficients of n

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32

Ramaswamy, Sivaraman, and Luis López-Bonilla Jose. "Combinatorial Game and Number Triangle." European Journal of Theoretical and Applied Sciences 1, no. 3 (2023): 50–54. https://doi.org/10.59324/ejtas.2023.1(3).05.

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By considering a triangular array of numbers for first six rows, we introduce a combinatorial game, whose solution depends on a number triangle. The conclusion brings us with a surprising consequence in deciding the result of the game. This paper analyzes the game and present the solution in detail using the number triangle which resembles the famous Leibniz Triangle of unit fractions.

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Lewis, Barry. "More power to Pascal." Mathematical Gazette 92, no. 525 (2008): 454–65. http://dx.doi.org/10.1017/s0025557200183652.

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Pascal’s triangle is the most famous of all number arrays - full of patterns and surprises. One surprise is the fact that lurking amongst these binomial coefficients are the triangular and pyramidal numbers of ancient Greece, the combinatorial numbers which arose in the Hindu studies of arrangements and selections, together with the Fibonacci numbers from medieval Italy. New identities continue to be discovered, so much so that their publication frequently excites no one but the discoverer.

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STĂNICĂ, Andreea-Amalia. "POLAND-LITHUANIA BINOMINAL AND THE EVOLUTION PROSPECTS OF THE LUBLIN TRIANGLE." INTERNATIONAL SCIENTIFIC CONFERINCE "STRATEGIESXXI" 18, no. 1 (2022): 117–21. http://dx.doi.org/10.53477/2971-8813-22-13.

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In the context of the annexation of Crimea, the need to counterbalance the political influence of the Russian Federation in the extreme central eastern region, severely affected by the manipulation of the euroatlantic base by the Soviet reminiscences, Poland, Lithuania and Ukraine laid the foundations of a military brigade and the Lublin Triangle. The strategic importance of cooperation formats is gaining new value in the context of the current aggression by the Russian Federation on the territory of Ukraine. The Lublin Triangle is based on historical congruences, in the first part of the rese

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35

Alitaneh, Saeed. "THEORIES ON COEFFICIENT OF VARIATION SCALES TRIANGLE AND NORMALIZATION OF DIFFERENT VARIABLES: A NEW MODEL IN DEVELOPMENT OF MULTIPLE CRITERIA DECISION ANALYSIS." International Journal of the Analytic Hierarchy Process 11, no. 2 (2019): 283–95. http://dx.doi.org/10.13033/ijahp.v11i2.565.

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This paper is an attempt to solve various problems by the two factors of mean and standard deviation (SD) of variables, introducing coefficient of variation (CV) of data as the best option for prioritization, scaling, pairwise comparison and normalization of quantitative and qualitative variables. An algorithm was built based on a coefficient of variation scales triangle (CVST) consisting of natural numbers with coefficients of binomial expansion for each line, followed by new and independent grading and scaling. In view of the existing factors, the theory provides higher generalization and ma

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36

Lagarias, Jeffrey C., and Harsh Mehta. "Products of binomial coefficients and unreduced Farey fractions." International Journal of Number Theory 12, no. 01 (2016): 57–91. http://dx.doi.org/10.1142/s1793042116500044.

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This paper studies the product [Formula: see text] of the binomial coefficients in the [Formula: see text]th row of Pascal’s triangle, which equals the reciprocal of the product of all the reduced and unreduced Farey fractions of order [Formula: see text]. It studies its size as a real number, measured by [Formula: see text], and its prime factorization, measured by the order of divisibility [Formula: see text] by a fixed prime [Formula: see text], each viewed as a function of [Formula: see text]. It derives three formulas for [Formula: see text], two of which relate it to base [Formula: see t

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37

Abramov, I. M., and G. S. Abramov. "GAUSS APPROXIMATION FOR NUMBER DISTRIBUTION IN OF A PASCAL’S TRIANGLE." Journal of Numerical and Applied Mathematics, no. 1 (135) (2021): 10–19. http://dx.doi.org/10.17721/2706-9699.2021.1.01.

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We received normal distribution parameters that approximates the distribution of numbers in the n-th row of Pascal's triangle. We calculated the values for normalized moments of even orders and shown their asymptotic tendency towards values corresponding to a normal distribution. We have received highly accurate approximations for central elements of even rows of Pascal's triangle, which allows for calculation of binomial, as well as trinomial (or, in general cases, multinomial) coefficients. A hypothesis is proposed, according to which it is possible that physical and physics-chemical process

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Strzałka, Dominik, Sławomir Wolski, and Andrzej Włoch. "Distance Fibonacci Polynomials by Graph Methods." Symmetry 13, no. 11 (2021): 2075. http://dx.doi.org/10.3390/sym13112075.

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In this paper we introduce and study a new generalization of Fibonacci polynomials which generalize Fibonacci, Jacobsthal and Narayana numbers, simultaneously. We give a graph interpretation of these polynomials and we obtain a binomial formula for them. Moreover by modification of Pascal’s triangle, which has a symmetric structure, we obtain matrices generated by coefficients of generalized Fibonacci polynomials. As a consequence, the direct formula for generalized Fibonacci polynomials was given. In addition, we determine matrix generators for generalized Fibonacci polynomials, using the sym

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de Alwis, Tilak. "Pascal type properties of Betti numbers." International Journal of Mathematics and Mathematical Sciences 17, no. 3 (1994): 545–52. http://dx.doi.org/10.1155/s0161171294000797.

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In this paper, we will describe the Pascal Type properties of Betti numbers of ideals associated ton-gons. These are quite similar to the properties enjoyed by the Pascal's Triangle, concerning the binomial coefficients. By definition, the Betti numbersβt(n)of an idealIassociated to ann-gon are the ranks of the modules in a free minimal resolution of theR-moduleR/I, whereRis the polynomial ringk[x1,x2,…,xn]. Herekis any field andx1,x2,…,xnare indeterminates. We will prove those properties using a specific formula for the Betti numbers.

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Hilton, Peter, and Jean Pedersen. "Relating Geometry and Algebra in the Pascal Triangle, Hexagon, Tetrahedron, and Cuboctahedron Part I: Binomial Coefficients, Extended Binomial Coefficients and Preparation for Further Work." College Mathematics Journal 30, no. 3 (1999): 170. http://dx.doi.org/10.2307/2687595.

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Hilton, Peter, and Jean Pedersen. "Relating Geometry and Algebra in the Pascal Triangle, Hexagon, Tetrahedron, and Cuboctahedron Part I: Binomial Coefficients, Extended Binomial Coefficients and Preparation for Further Work." College Mathematics Journal 30, no. 3 (1999): 170–86. http://dx.doi.org/10.1080/07468342.1999.11974053.

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Subbulakshmi, S. "Thirugnana Sambandhar - A Mathematician." Shanlax International Journal of Arts, Science and Humanities 9, no. 1 (2021): 136–40. http://dx.doi.org/10.34293/sijash.v9i1.3991.

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India has been the Land of notable poets whose exemplary works are world renowned. One such great poet is Thirugnana Sambandhar. He is a saint, poet, philosopher, composer who belongs to 7th Century. He was born in Seerkaazhi of Tamilnadu. He had coined many Special Geometrical poetic structures like Thiru ezhukkootrirukkai (poem with mathematical Triangular Pattern), Maalai Maatru (a poem with palindromic Structure), Mozhi Maatru (a poem in which the meaning of the poem can be observed by a systematic Chane of words), Gomuthri (Flow of the poem in such a way it forms a wave line), Chakramaatr

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Retnaningsih, Retnaningsih. "Fractal Geometry, Fibonacci Numbers, Golden Ratios, And Pascal Triangles as Designs." Journal of Academic Science 1, no. 1 (2024): 51–66. http://dx.doi.org/10.59613/msd4n328.

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Fractal geometry is a part of mathematics that discusses the shape of fractals or any form that is self-similarity. A fractal can be broken down into parts that are all similar to the original fractal. Fractals have infinite detail and can have self-similar structures at different magnifications. In many cases, a fractal can be generated by repeating a pattern, which is usually in a recursive or iterative process. In mathematics and art, two values are considered to be a golden ratio relationship if the ratio between the sum of the two values to the large value is equal to the ratio between th

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Krueger, Oliver, and Hans von Storch. "Evaluation of an Air Pressure–Based Proxy for Storm Activity." Journal of Climate 24, no. 10 (2011): 2612–19. http://dx.doi.org/10.1175/2011jcli3913.1.

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Abstract Yearly percentiles of geostrophic wind speeds serve as a widely used proxy for assessing past storm activity. Here, daily geostrophic wind speeds are derived from a geographical triangle of surface air pressure measurements and are used to build yearly frequency distributions. It is commonly believed, however unproven, that the variation of the statistics of strong geostrophic wind speeds describes the variation of statistics of ground-level wind speeds. This study evaluates this approach by examining the correlation between specific annual (seasonal) percentiles of geostrophic and of

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Ravaux, David Richard, Magin Zúñiga Estrada, and Erika Morales. "Factors Determining the Probability for Irregular Central American Migrants to Reside in the United States." Frontera norte 37 (January 1, 2025): 1–35. https://doi.org/10.33679/rfn.v1i1.2379.

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This article aims to study the factors that have favored the irregular entry and residence in the United States of migrants from the Northern Triangle of Central America during the period 2016-2019. The probabilities of success were estimated using data from Mexico’s Encuesta sobre Migración en la Frontera Sur (Southern Border Migration Survey), and a binomial logistic regression model was run to test the four hypotheses posed around the factors that determine the success of migrants in that journey. The results show that individual characteristics such as socio-demographic traits (country of

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GADJIEV, D. "METHODOLOGY OF THE TEACHING OF FINITE MATHEMATICS AT THE PEDAGOGICAL UNIVERSITIES DUE TO THE NEW TRENDS IN THE EVOLUTION OF THE MATHEMATICAL SCIENCES." World of academia: Culture, Education, no. 7 (October 15, 2021): 92–100. http://dx.doi.org/10.18522/2658-6983-2021-7-92-100.

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There were introduced new methods of the teaching and instruction of the following parts of the Pre-calculus: (1) Binomial Series; (2) Trigonometry; (3) Partial Fractions. The problems introduced in the article for the Pre-Calculus Course in Finite Mathematics was developed by the author. These unabridged problems are developed within the new trends in the evolutions of the novelty of the syllabi in Mathematics due to the development of the Mathematics Sciences / Theory and Applications. These new trends in the Theory and Application of Mathematics Sciences have been added new demands to the n

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Prabowo, Agung, Zulfatul Mukarromah, Lisnawati Lisnawati, and Pramono Sidi. "PENENTUAN HARGA OPSI BELI ATAS SAHAM PT. ANTAM (PERSERO) MENGGUNAKAN MODEL BINOMIAL FUZZY." Jurnal Matematika Sains dan Teknologi 19, no. 1 (2018): 8–24. http://dx.doi.org/10.33830/jmst.v19i1.124.2018.

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Option is a financial instrument where price depends on the underlying stock price. The pricing of options, both selling options and purchase options, may use the CRR (Cox-Ross-Rubinstein) binomial model. Only two possible parameters were used that is u if the stock price rises and d when the stock price down. One of the elements that determine option prices is volatility. In the binomial model CRR volatility is constant. In fact, the financial market price of stocks fluctuates so that volatility also fluctuates. This article discusses volatility of fluctuating stock price movements by modelin

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Rahmadewi, Yunda Maymanah, and Palupi Melati Pangastuti. "Karakteristik sensoris krimer kental dari kacang hijau menggunakan metode uji segitiga dan hedonik." Teknologi Pangan : Media Informasi dan Komunikasi Ilmiah Teknologi Pertanian 13, no. 1 (2022): 93–98. http://dx.doi.org/10.35891/tp.v13i1.2856.

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The shift in the vegan concept into a new lifestyle trend for urban communities has led to the development of plant-based foodstuffs into several processed products. Green beans with complex nutritional content have the potential to develop into a sweet thick creamer that can be developed as a complement to bakery pastry products. The purpose of this study was to develop sweetened condensed creamer made from mung bean as an alternative to filling in milk pies, to assess whether or not there was a difference between sweet thick creamer made from mung bean and animal sweetened condensed creamer,

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Zhang, Liyue, Kayla Buenaventura, Cassandra Wickstrom, Changqi Liu, and Mee Young Hong. "Sensory Evaluation of Blenderized Watermelon Flesh Juice With and Without the Rind." Current Developments in Nutrition 5, Supplement_2 (2021): 616. http://dx.doi.org/10.1093/cdn/nzab044_047.

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Abstract Objectives Although the rind of a watermelon contains much of health benefiting nutrients, it is much less appreciated by consumers and is often disposed. The purpose of this study was to compare sensory properties of blenderized watermelon (WM) flesh juice with and without rind. Methods Twenty-three semi-trained college students (age 26.1 ± 6.0 y, BMI 22.7 ± 3.1 kg/m2, 18 females and 5 males) participated in the sensory evaluation of blenderized WM flesh juice with and without the rind. Participants were asked to identify the unique sample among three samples in a triangle test. In a

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Burke, Maurice J., and Ted R. Hodgson. "Delving Deeper: Growth Rates and the Marvelous Geometric Sequence." Mathematics Teacher 103, no. 6 (2010): 458–62. http://dx.doi.org/10.5951/mt.103.6.0458.

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Students are often dazzled by the prodigious growth rate of the geometric sequence gn = 2n and the geometric series whose partial sums are Sn = 1 + 2 + 4 + 8 + … + 2n−1 = 2n − 1. Teachers sometimes note that the geometric sequence is the discrete “form” of an exponential function, which is characterized by very rapid growth. In particular, exponential functions grow faster than polynomial functions. A rigorous explanation of this claim is left to the calculus class in which students examine the relative growth rates of functions by using L'Hopital's rule. However, even by using tools developed

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