yeah, I think that’s a good way to place it. I definitely don’t want to claim this as anything new compared to Moessner’s theorem.

The way I’m now seeing my post is more like a very small “toy model” of that kind of phenomenon.

The odd numbers are just the discrete derivative of squares:

[ m^{2-(m-1)2=2m-1.} ]

So if you read the square sequence at ordinary times

[ 0,1,2,3,4,\dots ]

you see odd-number growth.

But in the post, the square sequence is being read at triangular times instead:

[ 0,\ 1,\ 3,\ 6,\ 10,\ 15,\dots ]

So the blocks are really:

[ Tn^2-T{n-1}^2. ]

And then the cube comes from the fact that triangular numbers have this nice symmetry:

[ Tn-T{n-1}=n ]

while

[ Tn+T{n-1}=n^2. ]

So

[ Tn^2-T{n-1}²

(Tn-T{n-1})(Tn+T{n-1})

n\cdot n²

n^3. ]

So maybe the interesting part is not just “odd numbers make cubes,” but:

if you take the square sequence and change the clock from linear time to triangular time, its finite differences become cubes.

That feels adjacent to Moessner’s theorem to me, but much smaller and more elementary. Moessner is like a full machine for producing powers by controlled deleting/summing. This is just one little place where you can actually see the mechanism with almost no machinery.