scholarly journals Computing Sticks against Random Walk

2019 ◽  
Vol 2 (1) ◽  
Keyword(s):  
Random Walk ◽  
Deterministic Model ◽  
Initial Conditions ◽  
Probabilistic Approach ◽  
Geometric Constructions ◽  
Pascal’S Triangle ◽  
Pascal Triangle ◽  
Usual Form ◽  
Recursive Formulas ◽  
Pascal's Triangle

A new deterministic model with the help of geometric constructions and computing sticks (not related to trajectories) is proposed for the new justification of consistency of the probabilistic approach to explain the random walk on a plane. A new, stepped form of the arithmetic triangle of Pascal based on the construction of horizontal and vertical lines (arrows) is suggested, a comparison is made with Pascal’s triangle of the usual form. A two-sided generalization of Pascal’s triangle is proposed. Geometric constructions and formulas for calculating the coefficients that fill in these new geometric (arithmetic) figures are given. Further types of generalization of the step-shaped Pascal triangle are proposed. Examples of generalized initial conditions and generalized recursive formulas for constructing various types of a generalized Pascal triangle are given.

scholarly journals PASCAL TRIANGLE AND PYTHAGOREAN TRIPLES

2021 ◽  
Vol 8 (8) ◽  
pp. 75-80
Author(s):  
R. Sivaraman
Keyword(s):  
Pythagorean Triples ◽  
Pascal’S Triangle ◽  
Pascal Triangle ◽  
Pascal's Triangle

The concept of Pascal’s triangle has fascinated mathematicians for several centuries. Similarly, the idea of Pythagorean triples prevailing for more than two millennia continue to surprise even today with its abundant properties and generalizations. In this paper, I have demonstrated ways through four theorems to determine Pythagorean triples using entries from Pascal’s triangle.

scholarly journals Distance Fibonacci Polynomials by Graph Methods

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2075
Author(s):  
Dominik Strzałka ◽  
Sławomir Wolski ◽  
Andrzej Włoch
Keyword(s):  
Initial Conditions ◽  
Symmetric Matrix ◽  
Fibonacci Polynomials ◽  
Pascal’S Triangle ◽  
Graph Interpretation ◽  
Symmetric Structure ◽  
Binomial Formula ◽  
Narayana Numbers ◽  
Pascal's Triangle

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 symmetric matrix of initial conditions.

RECURRENT TWO-DIMENSIONAL SEQUENCES GENERATED BY HOMOMORPHISMS OF FINITE ABELIAN p-GROUPS WITH PERIODIC INITIAL CONDITIONS

Fractals ◽  
2011 ◽  
Vol 19 (04) ◽  
pp. 431-442 ◽  
Author(s):  
MIHAI PRUNESCU
Keyword(s):  
Initial Conditions ◽  
Two Dimensional ◽  
Pascal’S Triangle ◽  
Modulo P ◽  
Pascal's Triangle ◽  
Context Free

We prove that if a recurrent two-dimensional sequence with periodic initial conditions coincides in a sufficiently large starting square with a two-dimensional sequence produced by an expansive system of context-free substitutions, then they must coincide everywhere. We apply this result for some examples built up by homomorphisms of finite abelian p-groups, in particular for Pascal's Triangle modulo pk, Pascal's Triangles modulo 2 with non-trivial periodic borders, and Sierpinski's Carpets with non-trivial periodic border. All these particular cases justify the conjecture that recurrent two-dimensional sequences generated by homomorphisms of finite abelian p-groups with periodic initial conditions can always be alternatively generated by expansive systems of context-free substitutions.

scholarly journals On Unimodality Problems in Pascal's Triangle

10.37236/837 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Xun-Tuan Su ◽  
Yi Wang
Keyword(s):  
Affirmative Answer ◽  
Binomial Coefficients ◽  
Pascal’S Triangle ◽  
Pascal Triangle ◽  
Pascal's Triangle

Many sequences of binomial coefficients share various unimodality properties. In this paper we consider the unimodality problem of a sequence of binomial coefficients located in a ray or a transversal of the Pascal triangle. Our results give in particular an affirmative answer to a conjecture of Belbachir et al which asserts that such a sequence of binomial coefficients must be unimodal. We also propose two more general conjectures.

Generalising Pascal's Triangle

2004 ◽  
Vol 88 (513) ◽  
pp. 447-456 ◽  
Author(s):  
Barry Lewis
Keyword(s):  
Prime Number ◽  
Recurrence Relation ◽  
Stirling Numbers ◽  
Binomial Coefficients ◽  
Combinatorial Properties ◽  
Pascal’S Triangle ◽  
Binomial Expansion ◽  
Pascal Triangle ◽  
Divisibility Property ◽  
Pascal's Triangle

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 exactly four of particular significance: those concerned with the Binomial coefficients, the Stirling numbers of both kinds, and a lesser known set of numbers – the Lah numbers. We also examine the combinatorial properties of the entries in these triangles and a prime number divisibility property that they all share. Thereby, we achieve a remarkable synthesis of these different entities.

Pascal's Prism

2012 ◽  
Vol 96 (536) ◽  
pp. 213-220
Author(s):  
Harlan J. Brothers
Keyword(s):  
Probability Theory ◽  
Fractal Geometry ◽  
Euclidean Geometry ◽  
Fibonacci Series ◽  
Pascal’S Triangle ◽  
Number Sequences ◽  
Definition Of ◽  
Image Position ◽  
Pascal's Triangle

Pascal's triangle is well known for its numerous connections to probability theory [1], combinatorics, Euclidean geometry, fractal geometry, and many number sequences including the Fibonacci series [2,3,4]. It also has a deep connection to the base of natural logarithms, e [5]. This link to e can be used as a springboard for generating a family of related triangles that together create a rich combinatoric object.2. From Pascal to LeibnizIn Brothers [5], the author shows that the growth of Pascal's triangle is related to the limit definition of e.Specifically, we define the sequence sn; as follows [6]:

3057. A Note on Pascal's Triangle

1963 ◽  
Vol 47 (359) ◽  
pp. 57
Author(s):  
Robert Croasdale
Keyword(s):  
Pascal’S Triangle ◽  
Pascal's Triangle
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