Kubic Hexagon Pascalloid Number Tiling Crystal

In a hexagonal number crystal, the first type of algorithm is just to add 6 numbers in a hexagonal fashion. The second type is to add 7 numbers, including the center number, as the process to make each number in the next iteration. I have worked out the constraints where the center variable is set equal to any given pair of opposite cornered variables multiplied by each-other. Now this third type of hexagon is to add 8 numbers, and uses the center number twice. It is the closed cube, and the other two hexagons may be a simple cube-ring polygon-based torroid at six variables half-opened cube at seven variables, and something represented by a completed Kubic numerical projection from three dimensions precisely aligned into two dimensions. Because it is arranged as thus the Combinatorics equation for this Cubic Hexagon may be found as



Download XL file:
Cubic Computation & Hexagonal Projection.xls

I have fetched this flattened cube or 8 variable hexagonal Pascalloid. Think of it as the hexagonal shadow of a cube when sunlight is hitting it at a certain angle.

You may be supprised to see it "as it is". The terms in this peculiar hexagon are soluable in algorithmic and a combinatoric and algebraic ways. Setting all the variables to '1', the terms drop out as usual, and it can remain a 2D recursive number pattern. And here is our algebraic pattern.







N = 0
N = 1
N = 2
N = 3
N = 4
N = 5
N = 6
N = 7
N = 8
N = 9
N = 10
N = 11
N = 12
N = 13
N = 14
N = 15
N = 16
N = 17
N = 18