4,10,20…
Hey so I was looking into the 4,10,20 thing. Someone mentioned that these were tetrahedral numbers, which are the sums of consecutive triangular numbers. Triangular numbers are just the sums of consecutive normal numbers, which are sums of consecutive ones. So I listed the numbers, then the triangular with respect to that number (n(n+1))/2. Then I did the tetrahedral with respect to that number (n(n+1)(n+2))/2. And I got this first pic. Then I noticed the block on the top had 4,10,20, and it was mirrored across the diagonal. So I decided to extrapolate the pattern and make this excel sheet.
Basically column one is a triangle in 0 dimensions (all 1's)
column 2 is a triangle in 1 dimension (just a line)
column 3 is a triangle in 2 dimensions (actual triangle)
column 4 is a triangle in 3 dimensions (pyramid)
then, column n is a triangle in n-1 dimensions.
Let T(n,x) = triangle in nth dimension for value x
You can flip these across the axis, so if you have some n-dimensional triangle for T(n,x), then you also know that it is T(x,n). I don't know I thought this was cool because it has to do with triangles of higher dimensions. Maybe we can use this to help us.