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!
I don't think I can help with The Big Picture but . . .
Many years ago, I played with multi-deimensional "triangular" numbers.
It was this time of year and a few classmates and I were drinking eggnog
. . and singing "The Twelve Days of Christmas".
We already knew about Triangular Numbers but someone asked,
. . "How many gifts were given during The Twelve Days?"
A little scribbling and we derived the formula for Tetrahedral Numbes
. . (a term we thought we invented): .
We had more eggnog and wondered about the sum of Tetrahedral numbers.
. . This turned out to be: .
We called them "Pentahedroidal" numbers.
In 4-space, this figure would be bounded by five tetrahedrons,
. . and would be called a pentahedroid.
We also wondered what scenario would produce this problem.
Suppose My True Love decided to spread the gift-giving over twelve years.
. . The first year, she gives me gift.
. . The second year, she gives me gifts.
. . The third year, gifts.
. . . . . . . . . . . . .
. . The twelfth year, gifts.
At this point we ran out of eggnog and brain cells . . .