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.
In this paper we deal with a method for the determination of numbers in a Pascal triangle that are simultaneously triangular, tetrahedral and pentaedroidni.
The collected results, obtained by mathematical analysis, were verified by computer. For this purpose, we used the C# programming language as well as the computer laboratory within our University in order to test the results. The results collected by computer confirmed the accuracy of the results obtained by mathematical analysis.
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.
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 .