Finally, there are the so-called Polygonal Numbers withinthe Pascal’s Triangle representing the sum of vertexes formed by polygons incertain figures. The prime polygon number is always 1.
Further, the secondnumber depends on the amount of vertexes in the polygon. The addition of allthe points and vertexes, as well as the extension of the polygon sides,produces the third polygonal number. At that, the vertexes are arranged tocomprise a larger polygon.
Square Numbers and TriangularNumbers are the following patterns within the Pascal’s Triangle that make up polygonalnumbers. The triangular numbers in the diagonal start from row 3 from thenumber one (1) to the number three (3), further to the number six (6), and tothe number ten (10), and so forth. Square numbers, in their turn, lie in thesame diagonal and make up the sum of the two numbers whenever the diagram has arounded area.
Square numbers start with the number 02, then goes 12, followedby 22, and then 32, respectively.Within the Pascal’s Triangle, Fibonnacci’s Sequence assumesthe sum of the numbers in the rows in consecutive sequence. The two consecutivenumbers are added in sequence to make the next number: 1,1,2,3,5,8,13,21,34,55,89,144,233. The Fibonacci sequence assumes that the set consequence of thetwo numbers of spirals.
The Fibonnacci Sequence is the curve featuring stringinstruments, in particular, a grand piano. The Fibonacci numbers are alsowidespread in natural forms, including spirals, flower petals, sunflower headseeds etc. Various spirals are arranged in either clockwise or counter-clockwisedirections. 1+6+21+56 = 84; 1+7+28+84+210+462+924 = 1716; 1+12 = 13.The ‘Hockey Stick Pattern’ assumes the diagonal arrangement ofnumbers starting with the ones (1’s) that border across the sides of the Triangleand end at any of the numbers on the same diagonal, the numbers inside theselection will coincide with the number below the end of the selection asfollows:20 = 1; 21 = 1+1 = 2; 22 = 1+2+1 = 4; 23 = 1+3+3+1 = 8; and 24= 1+4+6+4+1 = 16.
The second pattern in the Pascal’s Triangle is referred toas ‘The Sums of the Rows,’ meaning that the sum of the numbers, whatever therow is, always equals to 2 or ‘2n’ providing that ‘n’ features the number ofthe row. The pattern looks as follows: The first pattern in the Pascal’s Triangle is known as’Prime Numbers.’ Providing that the a prime number is the first element in arow, given that one (1) is the zero (0th) element of every row, thenall the numbers within the row are divided by it, save as the ones (1’s). Inrow 7, all the numbers from 7 to 35 are divided by 7. There is an infinite sequence of the rows within theTriangle. The whole construction assumes the number 1 at the tip. This isreferred to the zero (0) row.
Further, the first row is made by two numbers,namely ones (1’s) as a result of the addition of the two numbers rightwards andleftwards, namely 1 and 0. Importantly, the numbers outside the Triangle areall referred to zeros. Accordingly, the second row of the Triangle looks like:0+1=1; 1+1=2; 1+0=1. While the idea of the Triangle originally appeared as far asin ancient India, Persia and China, Blasé Pascal’s major contribution was toemphasise the value of all the containing patterns within the Triangle. Whileassessing the numbers in all possible varieties, Pascal applied the arithmetictriangle.
The patterns of the Pascal’s Triangle have significantly contributedto the development of the probability theory and the study of statistics.