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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)
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)
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.
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)
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)
A formula for the sum of the reciprocals of Heptagonal numbers has the form :
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
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
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)
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).
http://www.mathisfunforum.com/viewtopic.php?id=17853 66.238.111.50 ( talk) 05:04, 3 June 2014 (UTC) — Preceding unsigned comment added by 66.238.111.50 ( talk) 20:17, 13 July 2013 (UTC)
Polygonal Number Counting Function 46.115.40.137 ( talk) 12:48, 20 October 2012 (UTC)
no idea what these edits are all about — Preceding unsigned comment added by JKshaw ( talk • contribs)
→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)
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)
The following Wikimedia Commons file used on this page has been nominated for deletion:
Participate in the deletion discussion at the nomination page. — Community Tech bot ( talk) 22:57, 9 June 2019 (UTC)
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)
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:
![]() | This article is rated Start-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||
|
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)
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)
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.
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)
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)
A formula for the sum of the reciprocals of Heptagonal numbers has the form :
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
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
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)
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).
http://www.mathisfunforum.com/viewtopic.php?id=17853 66.238.111.50 ( talk) 05:04, 3 June 2014 (UTC) — Preceding unsigned comment added by 66.238.111.50 ( talk) 20:17, 13 July 2013 (UTC)
Polygonal Number Counting Function 46.115.40.137 ( talk) 12:48, 20 October 2012 (UTC)
no idea what these edits are all about — Preceding unsigned comment added by JKshaw ( talk • contribs)
→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)
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)
The following Wikimedia Commons file used on this page has been nominated for deletion:
Participate in the deletion discussion at the nomination page. — Community Tech bot ( talk) 22:57, 9 June 2019 (UTC)
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)
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: