Finally, there are the so-called Polygonal Numbers within

the Pascal’s Triangle representing the sum of vertexes formed by polygons in

certain figures. The prime polygon number is always 1. Further, the second

number depends on the amount of vertexes in the polygon. The addition of all

the points and vertexes, as well as the extension of the polygon sides,

produces the third polygonal number. At that, the vertexes are arranged to

comprise a larger polygon.

Square Numbers and Triangular

Numbers are the following patterns within the Pascal’s Triangle that make up polygonal

numbers. The triangular numbers in the diagonal start from row 3 from the

number one (1) to the number three (3), further to the number six (6), and to

the number ten (10), and so forth. Square numbers, in their turn, lie in the

same diagonal and make up the sum of the two numbers whenever the diagram has a

rounded area. Square numbers start with the number 02, then goes 12, followed

by 22, and then 32, respectively.

Within the Pascal’s Triangle, Fibonnacci’s Sequence assumes

the sum of the numbers in the rows in consecutive sequence. The two consecutive

numbers 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 the

two numbers of spirals. The Fibonnacci Sequence is the curve featuring string

instruments, in particular, a grand piano. The Fibonacci numbers are also

widespread in natural forms, including spirals, flower petals, sunflower head

seeds etc. Various spirals are arranged in either clockwise or counter-clockwise

directions.

1+6+21+56 = 84; 1+7+28+84+210+462+924 = 1716; 1+12 = 13.

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The ‘Hockey Stick Pattern’ assumes the diagonal arrangement of

numbers starting with the ones (1’s) that border across the sides of the Triangle

and end at any of the numbers on the same diagonal, the numbers inside the

selection will coincide with the number below the end of the selection as

follows:

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 to

as ‘The Sums of the Rows,’ meaning that the sum of the numbers, whatever the

row is, always equals to 2 or ‘2n’ providing that ‘n’ features the number of

the 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 a

row, given that one (1) is the zero (0th) element of every row, then

all the numbers within the row are divided by it, save as the ones (1’s). In

row 7, all the numbers from 7 to 35 are divided by 7.

There is an infinite sequence of the rows within the

Triangle. The whole construction assumes the number 1 at the tip. This is

referred 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 and

leftwards, namely 1 and 0. Importantly, the numbers outside the Triangle are

all 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 as

in ancient India, Persia and China, Blasé Pascal’s major contribution was to

emphasise the value of all the containing patterns within the Triangle. While

assessing the numbers in all possible varieties, Pascal applied the arithmetic

triangle. The patterns of the Pascal’s Triangle have significantly contributed

to the development of the probability theory and the study of statistics.