Talk:Polygonal number

message: 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)
thread_title:  The correct formula for Polygonal Numbers 

message: 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)
thread_title:  I have added some links to a video Podcast and they were remove 

message: 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)
replies: text: 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)
thread_title:  A possible connection 

message: 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)
replies: text: \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)

text: Do you have a link to this reference book? Robo37 (talk) 16:25, 13 June 2010 (UTC)
thread_title:  Heptagonal Formula 

message: http://www.mathisfunforum.com/viewtopic.php?id=1785366.238.111.50 (talk) 05:04, 3 June 2014 (UTC)
replies: text: Is this rigorous mathematics? 155.137.183.254 (talk) 16:19, 19 April 2023 (UTC)
thread_title:  Polygonal Number Counting Function 

message: 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)
thread_title:  What is going on with today's edits? Feb 7 

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

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

message: i do not understand why the article says it isnt. it would go: 
       ..
      ... 
.     ..     (1 then 2)
            ....
           .....
  ...     ......
 ....    .......  
.....    ......
....     .....
...      ....      (3 then 4)
this is totally possible! Bumpf (talk) 21:43, 10 April 2021 (UTC)
replies: text: Those are not hexagonal numbers of dots. Instead, they are the centered hexagonal numbers. —David Eppstein (talk) 22:15, 10 April 2021 (UTC)
thread_title:  Hexagons of dots could totally be a perfect lattice! 

message: 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)
thread_title:  Combinaisons