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.
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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.
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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:93∑11Hn≈1.30169996966629 and πtanhπ2√3√3≈1.50944975010619
Given that the last term in the sequence on the left is 1/25669 ≈ 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:93∑1Hn2n≈12.9999999999999999999999972942
Property 3
Notice that the summation begins at n=0 and this is confusing because H0 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):93∑1Hnn!=4e−1 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).
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Figure 3: http://www.numbersaplenty.com/set/heptagonal_number/ |
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Figure 4: http://www.numbersaplenty.com/set/octagonal_number/ |
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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 π and e, yet another example of the connectivity of numbers that I discussed in my previous post.
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