Pascals Triangle

What is Pascals Triangle?

The Pascals Triangle is an abstract mathematical structure achieved through adding numbers in a row together and putting the result below them on the next column . You start off with 1 and add the numbers next to it.

                            0
                          0 1 0
                         0 1 1 0
                        0 1 2 1 0
                       0 1 3 3 1 0
                      0 1 4 6 4 1 0
                    0 1 5 10 10 5 1 0
                   0 1 6 15 20 15 6 1 0
                 0 1 7 21 35 35 21 7 1 0 

There are many different patterns and relationships in the pascals triangle..

Note: For more information on the other patterns in pascals triangle, i recommend watching

=> This video on it by Numberphile

Pascals triangle mod 2

                           1 
                          1 1 
                         1[2]1
                        1 3 3 1
                      1[4][6][4]1
                     1 5[10][10] 5 1
                   1[6]15 [20] 15[6]1 
                  1 7 21 35 35 21  7 1

Its kind of hard to see with just ascii but if you look closely you will notice that the structure of numbers looks vaugely like the sierinski triangle and sure enough, yes it is.

I will do the triangle again, this time labeling numbers divisible by 2 "0" and those not divisible by 2 "1". This is what "mod 2" means really, instead of our base 10 numbers, its base 2.

                           1
                          1 1  
                         1 0 1
                        1 1 1 1
                       1 0 0 0 1
                      1 1 0 0 1 1
                     1 0 1 0 1 0 1
                    1 1 1 1 1 1 1 1

Can you see it?

                           1   row 1
                          1 1  row 2
                         1 0 1
                        1 1 1 1  row 4
                      -----------
                       1\0 0 0/ 1
                      1 1\0 0/ 1 1
                     1 0 1\0/ 1 0 1
                    1 1 1 1V 1 1 1 1 row 8

Thats right! Its the sierpinski triangle structure. Note that the rows filled with 1 act as the dividing lines, are exactly on powers of 2.