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 — 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.

• [1] 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.

This user is a recent changes patroller.

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

Assessment comment

The comment(s) below were originally left at Talk:Polygonal number/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

More wikilinks and references would help: needs to be more connected with other parts of wikipedia. Two of the current references are actually external links. Geometry guy 20:40, 31 May 2007 (UTC)

Last edited at 20:40, 31 May 2007 (UTC). 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:

• GrayDotX.svg

Participate in the deletion discussion at the nomination page. —Community Tech bot (talk) 22:57, 9 June 2019 (UTC)

Hexagons of dots could totally be a perfect lattice!

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)

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)

Parity of Polygonal Numbers

If the given polygon contains even number of sides, then the numbers alternate between odd and even. If the given polygon contains odd number of sides, then the numbers alternate between two odd and two even. 2409:40E0:F5:2AD1:ED51:6814:479B:38A5 (talk) 10:32, 31 October 2024 (UTC)