I have recently been investigating how number sequences relate to each other when put into different dimensions. The simplest example is squares, where the Nth term is T(n)=n^d (d=number of dimensions). So for a square (in two dimensions), the pattern is T(n)=n^2, so we get 1,4,9,16 etc. Then, when in three dimensions, the pattern is T(n)= n^3, so we get 1,8,27,64 etc. This obviously can go on for an infinite number of dimensions.
I then decided to try equilateral triangles. For triangular numbers, I worked out the pattern as T(n)=(n(n+1))/2. Then for tetrahedral numbers, I found that the pattern was T(n)=(n(n+1)(n+2))/6. Now this looks to me like if you add a dimension, you multiply the previous Nth term rule by (n+d-1)/d. However, I have no idea how to write this whole calculation! The furthest I can get is (n...whatever I'm missing...)/d! . Please Help!