Talk:Polygonal number

Incorrect formula
I think that the correct formula for Polygonal Numbers is p(a,s)=[(a-2)s-(a-4)]s/2 where 'a' represent the number of angles and 's' the length of the sides of a regular polygon.

In the Wikipedia's text it seems that there is a mistake in it. Check it!

Raul Nunes - (raulnunes@threebirds.com) — Preceding unsigned comment added by 200.169.137.13 (talk) 09:06, 9 September 2003 (UTC)


 * Looks like you're right &mdash; the formula as stated for s-polygonal numbers doesn't match up with the formulas given for specific cases. I've corrected it. Factitious 11:50, Nov 15, 2004 (UTC)

The correct formula for Polygonal Numbers
OK! It seems all-correct now! Raul Nunes (raulnunes@threebirds.com.br) — Preceding unsigned comment added by 201.1.64.134 (talk) 20:04, 28 November 2004 (UTC)

I have added some links to a video Podcast and they were remove
Hi everyone I have added some links to a video podcast that I own. I think they are a nice addition to wikipedia please look at them and express you oppinion here , judge for yourself if the links are really useful or not to wikipedia.


 * Video PodCast by http://www.isallaboutmath.com/

If any of you think they are valuable to wikipedia then feel free to add them back in the external links.

Regards SilentVoice 03:23, 22 January 2007 (UTC)

A possible connection
Correct me if I'm wrong, but aren't polygonal numbers connected in some manner to Arithmetic series? --202.139.5.61 (talk) 12:04, 11 November 2008 (UTC)


 * They can be connected. The nth (s+2)-gonal number is the sum of the first n terms of the arithmetic progression 1, 1+s, 1+2s, 1+3s, ... PrimeHunter (talk) 22:26, 11 November 2008 (UTC)

Heptagonal Formula
A formula for the sum of the reciprocals of Heptagonal numbers has the form :
 * $$ \sum_{n=1}^\infty \frac{2}{n(5n-3)} = \frac{1}{15}{\pi}{\sqrt{25-10\sqrt{5}}}+\frac{2}{3}\ln(5)+\frac{{1}+\sqrt{5}}{3}\ln\left(\frac{1}{2}\sqrt{10-2\sqrt{5}}\right)+\frac{{1}-\sqrt{5}}{3}\ln\left(\frac{1}{2}\sqrt{10+2\sqrt{5}}\right)

$$ It might be interesting to add an additional example, in this case , the the sum of the reciprocals of Heptagonal numbers. User:Alanonala http://en.wikipedia.org/wiki/User:Alanonala
 * Cool, how did you work that out? Have you got a ref, and do you know the sum of reciprocals of any other Polygonal numbers? It might be a good idea to try to get results for every Polygonal number on the table. Robo37 (talk) 17:23, 22 May 2010 (UTC)

Basically the digamma function of Gauss,( see Digamma function ). For positive integers m and k (with m < k), the digamma function may be expressed in terms of elementary functions as


 * $$\Psi\left(\frac{m}{k}\right) = -\gamma -\log(2k)

-\frac{\pi}{2}\cot\left(\frac{m\pi}{k}\right) +2\sum_{n=1}^{\lceil (k-1)/2\rceil} \cos\left(\frac{2\pi nm}{k} \right) \log\left(\sin\left(\frac{n\pi}{k}\right)\right) $$ You actually need gamma on the left side of the equation. That's actually really lucky! The series you need to sum is: S = - (2/(r - 4) [ gamma + digamma(2/(r - 2)) ].  Where r is the number of sides of the polygon, gamma is the Euler-Mascheroni constant, digamma is as before. The only thing I've added is the reduction of value of sin(pi*p/5),cos(pi*p/5), cot(pi*p/5) to the correct values using square roots and so on. Here p is usually 1,2,4,8 so it's possible to work out the formula for the Heptagonal numbers. Alanonala

One more thing, the sigma in the formula is divided by two if the number is even. This makes the even sided polygons half as much work. Oh, and let me just add that Gauss is doing all the heavy lifting, (as you'd expect).Alanonala —Preceding undated comment added 17:52, 24 May 2010 (UTC).

My reference book on Special Functions has log(k) in place of log(2k) in the above formula. The top summand is also slightly different floor(k/2) in place of ceiling((k-1)/2) Alanonala —Preceding undated comment added 18:15, 24 May 2010 (UTC).

5   1.482037501770111 6    1.386294361119891 7    1.322779253122389 8    1.277409057559637 9    1.243320926153713 10   1.216745956158244 11   1.195434116529628 12   1.177956057922664 13   1.163358901106344 14   1.150982368094676 15   1.140354178594879 16   1.131127429553802 17   1.12304149271332 18   1.115896714056332 19   1.109537538129369 20   1.103840951528787 21   1.098708384537068 22   1.09405991950106 23   1.089830073270757 24   1.085964675709217 25   1.082418525428977 26   1.079153605775087 27   1.076137710704363 28   1.073343374655556 29   1.070747030685862 30   1.068328341980459

some approximate values for r=5 to 30 using the equation: S(r) = - (2/(r - 4) [ gamma + digamma(2/(r - 2)) ]Alanonala —Preceding unsigned comment added by 96.40.190.218 (talk) 23:30, 24 May 2010 (UTC)
 * Do you have a link to this reference book? Robo37 (talk) 16:25, 13 June 2010 (UTC)

I was very curious about the digamma function, so I looked up Gauss' article, "Circa Seriem Infinitam ..." Gauss Werke Vol. 3, page 157, article 74 and 75. The formula for digamma in terms of simple functions is listed twice as it differs by case, (even versus odd denominator). There is an additional log(2) added to the formula in article 75. It yields the correct value for odd denominator, otherwise we would have log(10) rather than log(5) in the formula for r = 7, which is incorrect. Thanks for putting up a proper reference that includes r=7 in the body of the article. Alanonala (talk) —Preceding undated comment added 19:11, 16 June 2010 (UTC).

Sorry, vice versa for the even or odd case in the above. Alanonala (talk) —Preceding undated comment added 20:01, 17 June 2010 (UTC).

Polygonal Number Counting Function
http://www.mathisfunforum.com/viewtopic.php?id=1785366.238.111.50 (talk) 05:04, 3 June 2014 (UTC) — Preceding unsigned comment added by 66.238.111.50 (talk) 20:17, 13 July 2013 (UTC)


 * Is this rigorous mathematics? 155.137.183.254 (talk) 16:19, 19 April 2023 (UTC)

Polygonal Number Counting Function 46.115.40.137 (talk) 12:48, 20 October 2012 (UTC)

What is going on with today's edits? Feb 7
no idea what these edits are all about — Preceding unsigned comment added by JKshaw (talk • contribs)
 * Huh? The only edits to the article today involved someone adding some unnecessarily large numbers to the tables of numbers, and being reverted. What is worthy of comment in that? —David Eppstein (talk) 22:13, 7 February 2015 (UTC)

→Just wanted to bring it to the attention of someone who knows the mathematics behind it.

JKshaw (talk) 22:21, 7 February 2015 (UTC)

Assessment comment
Substituted at 02:29, 5 May 2016 (UTC)

A Commons file used on this page has been nominated for deletion
The following Wikimedia Commons file used on this page has been nominated for deletion: Participate in the deletion discussion at the. —Community Tech bot (talk) 22:57, 9 June 2019 (UTC)
 * GrayDotX.svg

Hexagons of dots could totally be a perfect lattice!
i do not understand why the article says it isnt. it would go:

this is totally possible! Bumpf (talk) 21:43, 10 April 2021 (UTC)
 * Those are not hexagonal numbers of dots. Instead, they are the centered hexagonal numbers. —David Eppstein (talk) 22:15, 10 April 2021 (UTC)

Combinaisons
in the Combinations section, the article mentions: "The number 1225 is hecatonicositetragonal (s = 124), hexacontagonal (s = 60), icosienneagonal (s = 29), hexagonal, square, and triangular.". That is true. But it's just kind of arbitrary and just the first one that appears in 6 sets other than the natural number set (2-gonals) and its own set (1225-gonal). Other numbers (including 1540, 2926, 4005, 5985, 8856, etc...) also appear in 6 sets. Furthermore, other numbers such as:
 * 11781 appear in 8 sets
 * 27405 appear in 9 sets
 * 220780 appear in 10 sets
 * 203841 appear in 11 sets...
 * So again, this appear to be kind of arbitrary, and is it notable?
 * Also, 1 isn't the only number that appear in all sets, 0 also appear in all sets, even though is it sometimes omitted in a polygonal sequence, and it can be argued that it should or shouldn't be there.

Dhrm77 (talk) 19:53, 13 November 2023 (UTC)