Pascal's TriangleAnd Its Patterns 
At the tip of Pascal's Triangle is the number 1, which makes up the
0^{th} row. The first row (1 & 1) contains two 1's, both formed
by adding the two numbers above them to the left and the right, in this
case 1 and 0. All numbers outside the triangle are considered 0's. Do the
same to create the 2nd row: 0+1=1; 1+1=2; 1+0=1. And
the third: 0+1=1; 1+2=3; 2+1=3; 1+0=1. In this
way, the rows of the triangle go on forever. A number in the triangle can
also be found by nCr (n Choose r) where n is the number of the row and
r is the element in that row. For example, in row 3, 1 is the 0^{th}
element, 3 is first element, the next three is the 2^{nd} element,
and the last 1 is the 3^{rd} element. The formula for nCr is:


The sum of the numbers in any row is equal to 2 to the
n^{th
}power or 2^{n}, when n is the number of the row.
For example:
2^{0} = 1
2^{1} = 1+1 = 2 2^{2} = 1+2+1 = 4 2^{3} = 1+3+3+1 = 8 2^{4} = 1+4+6+4+1 = 16 
If the first element in a row is a prime number (remember, the 0th element of every row is 1), all the numbers in that row (excluding the 1's) are divisible by it. For example, in row 7 (1 7 21 35 35 21 7 1) 7, 21, and 35 are all divisible by 7.
If a diagonal of numbers of any length is selected starting at any
of the 1's bordering the sides of the triangle and ending on any number
inside the triangle on that diagonal, the sum of the numbers inside the
selection is equal to the number below the end of the selection that is
not on the same diagonal itself. If you don't understand that, look at
the drawing.
1+6+21+56 = 84

If a row is made into a single number by using each element as a digit of the number (carrying over when an element itself has more than one digit), the number is equal to 11 to the n^{th }power or 11^{n} when n is the number of the row the multidigit number was taken from.
Row #  Formula  =  MultiDigit number  Actual Row 
Row 0  11^{0}  =  1  1 
Row 1  11^{1}  =  11  1 1 
Row 2  11^{2}  =  121  1 2 1 
Row 3  11^{3}  =  1331  1 3 3 1 
Row 4  11^{4}  =  14641  1 4 6 4 1 
Row 5  11^{5}  =  161051  1 5 10 10 5 1 
Row 6  11^{6}  =  1771561  1 6 15 20 15 6 1 
Row 7  11^{7}  =  19487171  1 7 21 35 35 21 7 1 
Row 8  11^{8}  =  214358881  1 8 28 56 70 56 28 8 1 
Fibonacci's sequence can also be located in Pascal's Triangle. The
sum of the numbers in the consecutive rows shown in the diagram are the
first numbers of the Fibonacci Sequence. The sequence can also be formed
in a more direct way, very similar to the method used to form the triangle,
by adding two consecutive numbers in the sequence to produce the next number.
The creates the sequence: 1,1,2,3,5,8,13,21,34, 55,89,144,233, etc . .
. . The Fibonacci Sequence can be found in the golden rectangle, the lengths
of the segments of a pentagram, and in nature, and it decribes a curve
which can be found in string instruments, such as the curve of a grand
piano. The formula for the n^{th} term in
the Fibonacci Sequence is

Triangular Numbers are just one type of Polygonal Number. See the section on Polygonal Numbers for an explanation. The Triangular Numbers can be found in the diagonal starting at row 3 as shown in the diagram. The first Triangular Number is 1, the second is 3, the third is 6, the fourth is 10, and so on. 
Square Numbers are another type of Polygonal Number. They are found in the same diagonal as the Triangular Numbers. A Square Number is the sum of the two numbers in any circled area in the diagram. (The colors are different only to distinguish between the separate "rubber bands"). The n^{th} Square Number is equal to the n^{th} Triangular Number plus the (n1)^{th} Triangular Number. (Remember, any number outside the triangle is 0). The interesting thing about these 4sided Polygonal Numbers is that their name explains them perfectly. The very first Square Number is 0^{2}. The second is 1^{2}, the third is 2^{2} (4), the fourth is 3^{2} (9), and so on. Read on to learn more. 
Polygonal Numbers are really just the number of vertexes in a figure formed by a certain polygon. The first number in any group of Polygonal Numbers is always 1, or a point. The second number is equal to the number of vertexes of the polygon. For example, the second Pentagonal Number is 5, since pentagons have 5 vertexes (and sides). The third Polygonal Number is made by extending two of the sides of the polygon from the second Polygonal Number, completing the larger polygon, and placing vertexes and other points where necessary. The third Polygonal Number is found by adding all the vertexes and points in the resulting figure. (Look at the table below for a clearer explaination). A formula that will generate the n^{th }xgonal number (for example: the 2^{nd} 3gonal, or triangular number) is:
Type  1^{st}  2^{nd}  3^{rd}  4^{th}  5^{th}  6^{th} 
Triangular  
Value  1  3  6  10  15  21 
Square  
Value  1  4  9  16  25  36 
Pentagonal  
Value  1  5  12  22  35  51 
Hexagonal  
Value  1  6  15  28  45  66 
Image  Points  Segments  Triangles  Quadrilaterals  Pentagons  Hexagons  Heptagons 
1  
2  1  
3  3  1  
4  6  4  1  
5  10  10  5  1  
6  15  20  15  6  1  
7  21  35  35  21  7  1 
As you may have noticed, the numbers in the chart above are actually the tip of the rightangled form of Pascal's Triangle, except the preceeding 1's in each row are missing. The circular figures are formed by simply placing a number of points on a circle and then drawing all the possible lines between them. This chart shows that for a figure with n points, all you need to do is look at the n^{th }row of the triangle in order to find the number of points, line segments, and polygons in the figure with ALL of their vertices on the circle.
When all the odd numbers (numbers not divisible by 2) in Pascal's Triangle are filled in (black) and the rest (the evens) are left blank (white), the recursive Sierpinski Triangle fractal is revealed (see figure at right), showing yet another pattern in Pascal's Triangle. Other interesting patterns are formed if the elements not divisible by other numbers are filled, especially those indivisible by prime numbers. Download and decompass the program Pascal's Triangle, then give it a whirl! No installation required. The program will allow you to reveal/color the multiples of 2 to 24 inclusive in the first 20 rows of Pascals's Triangle and to view a zoom of the pattern continued for the first 128 rows. 
Source: http://ptri1.tripod.com/