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 $3n(n-1)+1$ 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.