>
Pascal's TriangleAnd Its Patterns 
At the tip
of Pascal's Triangle is the number 1, which makes up the zeroth 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 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 infinitly. 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 zeroth element, 3 is element number 1, the next three is the 2nd element, and
the last 1 is the 3rd element. The formula for nCr is:  r!(nr)! 

! means factorial, or the preceeding number multiplied by all the positive integers that are smaller than the number. 5! = 5 × 4 × 3 × 2 × 1 = 120. 
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 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 1+7+28+84+210+462+924 = 1716 1+12 = 13 
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 
Fibonnacci'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 Fibonnacci
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 Fibonnacci 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} number
in the Fibonnacci Sequence is
Try the formula out below (requires JavaScript) 
Triangular Numbers are just one type of polygonal numbers. See the section on Polygonal Numbers for an explaination of polygonal and triangular numbers. 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 Numbers 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 the Polygonal Number section to learn more. 
Type  1^{st}  2^{nd}  3^{rd}  4^{th}  5^{th}  6^{th}  7^{th} 
Triangular  
Value  1  3  6  10  15  21  28 
Square  
Value  1  4  9  16  25  36  49 
Pentagonal  
Value  1  5  12  22  35  51  70 
Hexagonal  
Value  1  6  15  28  45  66  91 
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 
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 near 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. Go here to download programs that calculate Pascal's Triangle and then use it to create patterns, such as the detailed, rightangle Sierpinski Triangle at the far right. 