Explore the vast range of titles available.

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 triangle consisting of integers which are congruent to 1 modulo 4 except the first row. In this application, we use the congruence ( x + 1 ) p e ≡ ( x p + 1 ) p e − 1 ( mod p e ) of binomial expansions which we could prove for any prime number p and any positive integer e. We think that this article is fit for the Special Issue “Number Theory and Symmetry,” since we prove a symmetric property on the 4-Pascal’s triangle by means of a number-theoretical property of binomial expansions.

In this paper, we describe a method to determine the power sum of the elements of the rows in the hyperbolic Pascal triangles corresponding to –4, q} with q ≥ 5. The method is based on the theory of linear recurrences, and the results are demonstrated by evaluating the kth power sum in the range 2 ≤ k ≤ 11.

This paper introduces a novel code called two dimensional Pascal’s triangle zero cross correlation (2D-PTZCC) for spectral/spatial coding with its structure of the corresponding system to implement in spectral amplitude coding optical code division multiple access (SAC-OCDMA) systems The novel code is derived from a one dimensional zero cross correlation (1D-ZCC) code using Pascal’s triangle rule. The analytical results prove that the proposed code has totally removed the multiple access interference (MAI), in addition the phase induced intensity noise (PIIN) influence is highly reduced due to the zero cross correlation (ZCC) property. Comparing with the recent developed two-dimensional codes like 2D multi diagonal (2D-MD), 2D dynamic cyclic shift (2D-DCS), 2D diluted perfect difference (2D-DPD) and 2D perfect difference (2D-PD) codes for the same code length. The results of simulation show that the suggested code improve the system capacity and increase the number of simultaneous users reaches 29 % and 64 % comparing to 2D-DPD and 2D-PD codes, respectively. As well, it supplies higher signal power and data bit rates whereas it saves effective source power around −0.81dBm for 2D-DCS and 2D-DPD codes and consumes minor light spectral width. The passage from 1D to 2D ameliorated 1.68 times the system capacity and saved around the effective power for each user.

30 pages A binary triangle of size$n$is a triangle of zeroes and ones, with$n$rows, built with the same local rule as the standard Pascal triangle modulo$2$ . A binary triangle is said to be balanced if the absolute difference between the numbers of zeroes and ones that constitute this triangle is at most$1$ . In this paper, the existence of balanced binary triangles of size$n$ , for all positive integers$n$ , is shown. This is achieved by considering periodic balanced binary triangles, that are balanced binary triangles where each row, column or diagonal is a periodic sequence.

We survey classical results and recent developments, old and new problems, conjectures and ideas selected from the endless theme of iterated application of the fundamental rule of addition. The multitude of forms created by this spring, like the commandment \"Let there be light\" from the first day of creation, is emphasized by the role played by the prime numbers. The subject sounds harmoniously between poetry and astronomy or geometry, finding its origin in the East at Pingala (4-2nd century BC) and in the West at Apollonius of Perga (3rd century BC). Our work is divided in two parts: the present paper is mostly dedicated to playing with numbers, while the second one, which will follow in a companion paper, is based on geometrical motives.

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.