Pascal's Triangle

There are many patterns in Pascal's Triangle. One of the most visible is the powers of 11. It goes like so:
1=11^0
11=11^1
1 2 1=11^2
1 3 3 1=11^3
1 4 6 4 1=11^4
1 5 10 10 5 1=11^5
1 6 15 20 15 6 1=11^6
1 7 21 35 35 21 7 1=11^7

So we all know that 11^2 is 121 but once the numbers start getting to the double digits like 1 7 21 and such we have to carry number over in order to get the correct answer. 11^7 is really 19487171. In order to get this answer from Pascal's Triangle we carry all of the tens to the next column. So from right to left- 1 7 1, now add the 2 from 21 to 5, 7, now add the 3 from 35 to 5, 8, now add the 3 from 35 to 1, 4, now add the 2 from 21 to 7, 9 1. 1 7 1 7 8 4 9 1. There we go!

Now for the other patterns in Pascal's Triangle. The sum of any given row is 2 to the rows number power. Example: Row 1= 1+1= 2^1, Row 2 = 1+2+1=2^2, so on and so forth.

Another pattern in the triangle that we have is Fibonacci's number. The pattern goes 1,1,2,3,5,8,13,21 you take the number you are given and at it to the last number to get your new number. In Pascal's Triangle if you make a sorta angled diagonal we get this pattern. First diagonal is 1. Second diagonal is 1. Third diagonal is 1+1=2. Fourth diagonal is 1+2=3. Continue.

There are probably many, many more patterns in Pascal's Triangle we just have to be willing to work it out far enough!