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.

Constants Associated with Centered Hexagonal and Other Figurate Numbers

Today I turned 25669 days old and this number happens to be the 93rd centered hexagonal number. These are figurate numbers because they can be represented as hexagonal rings surrounding a central dot. See Figure 1.

Figure 1

Centered hexagonal numbers can be written in the form \(3 \,n \, (n-1)+1\) where \(n=0,1,2,3, ... \). What struck me as interesting were some of the properties of this series which begins: 1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, 469, 547, 631, 721, 817, 919, 1027, 1141, 1261, ... I'm thankful to NumbersAplenty for alerting me to these properties. See Figure 2.

Figure 2: http://www.numbersaplenty.com/set/hex_number/

These results are quite amazing. The property on the left in Figure 2, I'll designate as property 1, the one in the middle as property 2 and the one on the right as property 3.

Property 1

Let's consider the sequence up to the 93rd hexagonal number and see how the results compare:$$ \sum_1^{93}{\frac{1}{H_n}} \approx 1.30169996966629 \text{ and } \frac{\pi \,\tanh{\frac{\pi}{2\sqrt{3}}}}{\sqrt{3}} \approx 1.50944975010619$$Given that the last term in the sequence on the left is 1/25669 \( \approx \) 0.0000389574973703689, there are clearly a great many more terms to add before the figure on the right is approached.

Property 2

By contrast, this sequence approaches 13 fairly rapidly:$$ \sum_1^{93} \frac{H_n}{2^n} \approx 12.9999999999999999999999972942$$Property 3

Notice that the summation begins at \(n=0\) and this is confusing because \(H_0\) is not really defined. It might be better to begin the summation at \(n=1\) and make the right side equal to \(4e-1\). In this case, the summation on the left again rapidly approaches the value on the right (not surprising in view of the factorial in the denominator):$$ \sum_1^{93} \frac{H_n}{n!} = 4e -1\text{ to at least 100 decimal places}$$There are interesting results for the sums of the reciprocals of other figurate numbers. Some of these are shown in Figure 3 (heptagonal), Figure 4 (octagonal) and Figure 5 (decagonal).

Figure 3: http://www.numbersaplenty.com/set/heptagonal_number/

Figure 4: http://www.numbersaplenty.com/set/octagonal_number/

Figure 5: http://www.numbersaplenty.com/set/decagonal_number/

There are plenty more but that's enough for the moment. In the meantime, the centered hexagonal and other figurate numbers have been shown to be linked to the mathematical constants \(\pi \) and \(e\), yet another example of the connectivity of numbers that I discussed in my previous post.