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.

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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.

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.

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