Bill's Triangle
Bill's Triangle is a triangle similar to Pascal's Triangle, except each number is obtained by adding the top three numbers, not just the top two. It was found by Bill9000. (If anyone finds someone who found this triangle before, please tell Bill9000.)
How to Make Bill's Triangle
As with Pascal's Triangle, we start with a complete row of zeros. However, in Bill's Triangle, we need another row:
![\[0, 0\]](http://latex.artofproblemsolving.com/5/e/0/5e0ede1941f205dbbfa24d2c2d2b72d70abfd655.png)
![\[0, 0, 0\]](http://latex.artofproblemsolving.com/3/0/a/30afeff78a22a6b76d6fd3a64a2f0b6982221c19.png)
Again, as with Pascal's Triangle, we change a zero to a one:
![\[0, 1, 0\]](http://latex.artofproblemsolving.com/1/b/5/1b5fecf24cfd67990adb180ea7a68817d2beb207.png)
Now, we can add the top three numbers to get the next number. The boxed numbers are the ones we add to get the next row.
![\[\boxed{0}, 0\]](http://latex.artofproblemsolving.com/c/4/1/c412641f322e031bdb64a91c0b9bd8d7d59d9aad.png)
![\[\boxed{0}, \boxed{1}, 0\]](http://latex.artofproblemsolving.com/7/e/b/7eb6639f87dc9e711f37e80dab29f4966c6a643a.png)
We add 
so that is the next number. We also add these numbers:
![\[0, \boxed{0}\]](http://latex.artofproblemsolving.com/c/c/f/ccfe4d54167bf88457d0d424578eba2c455c4a13.png)
![\[0, \boxed{1}, \boxed{0}\]](http://latex.artofproblemsolving.com/1/8/8/18811a346506fa511594c90a102e206da0616695.png)
Therefore, the first few rows of Bill's Triangle are
![\[1, 1\]](http://latex.artofproblemsolving.com/4/e/6/4e6a1628e7ebc304d8cb754b1236e42eef6b80e3.png)
