Polygonal Prism Number Confusion

Despite the fact that I've encountered figurate numbers many times during my research, I found myself getting confused by them and so this post is meant to clarify and document my understanding. Even though the title of this post relates to 3-dimensional prisms, I'm going to start in two dimensions with octagonal numbers. Figure 1 shows how the successive octagonal numbers are generated.

Figure 1

The formula associated with this visual process is \(3n^2-2n=n(3n-2)\) which yields the so-called octagonal numbers that form OEIS A000567:

A000567

Octagonal numbers: \(n(3n-2)\). Also called star numbers.

The initial members of the sequence are:

0, 1, 8, 21, 40, 65, 96, 133, 176, 225, 280, 341, 408, 481, 560, 645, 736, 833, 936, 1045, 1160, 1281, 1408, 1541, 1680, 1825, 1976, 2133, 2296, 2465, 2640, 2821, 3008, 3201, 3400, 3605, 3816, 4033, 4256, 4485, 4720, 4961, 5208, 5461

Now we can move on to three dimensions and consider the octagonal prism shown in Figure 2.

Figure 2

The octagonal prism numbers are formed simply by adding successive layers to the octagonal numbers as evidenced by the formula \(3n^3-2n^2=n(3n^2-2n)\) where we can see the octagonal number is multiplied by the number of layers. These numbers form OEIS A100176:

A100176

Structured octagonal prism numbers.

The initial members of the sequence are:

1, 16, 63, 160, 325, 576, 931, 1408, 2025, 2800, 3751, 4896, 6253, 7840, 9675, 11776, 14161, 16848, 19855, 23200, 26901, 30976, 35443, 40320, 45625, 51376, 57591, 64288, 71485, 79200, 87451, 96256, 105633, 115600, 126175, 137376, 149221

Thus it would seem that all the prism numbers can be calculated simply by multiplying the related two dimensional figurate number by the number of layers. Thus the formula for the hexagonal numbers is \( 2n^2-n \) and thus the formula for the hexagonal prism numbers would be \( n(2n^2-n)\). Oops, not so fast!

The formula for hexagonal prism numbers is \((n + 1)(3n^2 + 3n + 1) \) as stated in OEIS A005915. So what's going on. The problem would seem to lie in the use of the term "structured". OEIS A100176 refers to "structured" octagonal prism numbers and not octagonal prism numbers. In fact, a search of structured hexagonal prism numbers brings up OEIS A015237 whose members do conform with the formula \( n(2n^2-n)\).

This leads to the obvious question. What is the difference between a structured hexagonal or octagonal prism and a simple hexagonal or octagonal prism? So far I've not been able to find an answer to what seems like a straightforward question. I'll keep investigating. Meanwhile the concept of a structured polygonal prism and the numbers associated with it are easy enough to understand.

The general formula for the \(n\)-th polygonal number in a polygon of with \(s\) sides is:$$P(s,n)=\frac {(s-2)n^2-(s-4)n}{2}$$Checking this out we see that when \(s=6\):$$ \begin{align} P(6,n)&=\frac {4n^2-2n}{2}\\&=2n^2-n \end{align}$$which checks out with what was shown earlier. When \(s=8\), we have:$$ \begin{align} P(8,n)&=\frac {6n^2-4n}{2}\\&=3n^2-2n \end{align}$$which again checks out. Thus the general formula for \(n\)-th structured polygonal prism number in a prism with cross-section polygon of \(s\) sides is:$$ PP(s,n)=\frac {(s-2)n^3-(s-4)n^2}{2}$$So let's calculate the structured duodecagonal prism number when substituting \(s=12\). The result is:

1, 24, 99, 256, 525, 936, 1519, 2304, 3321, 4600, 6171, 8064, 10309, 12936, 15975, 19456, 23409, 27864, 32851, 38400, 44541

This is a sequence not listed in the OEIS and so it shall remain.