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In [[mathematics]], a '''polygonal number''' is a [[number]] represented  as dots or pebbles arranged in the shape of a [[regular polygon]]. The dots are thought of as alphas (units). These are one type of 2-dimensional [[figurate number]]s.
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


== Definition and examples ==
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
* Only registered users will be able to execute this rendering mode.
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.


The number 10, for example, can be arranged as a [[triangle]] (see [[triangular number]]):
Registered users will be able to choose between the following three rendering modes:  


:{|
'''MathML'''
| align="center" | [[Image:GrayDotX.svg|16px|*]]<br>[[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]]<br>[[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]]<br>[[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]]
:<math forcemathmode="mathml">E=mc^2</math>
|}


But 10 cannot be arranged as a [[square (geometry)|square]]. The number 9, on the other hand, can be (see [[square number]]):
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


:{|
'''source'''
| align="center" | [[Image:GrayDot.svg|16px|*]][[Image:GrayDot.svg|16px|*]][[Image:GrayDot.svg|16px|*]]<br>[[Image:GrayDot.svg|16px|*]][[Image:GrayDot.svg|16px|*]][[Image:GrayDot.svg|16px|*]]<br>[[Image:GrayDot.svg|16px|*]][[Image:GrayDot.svg|16px|*]][[Image:GrayDot.svg|16px|*]]
:<math forcemathmode="source">E=mc^2</math> -->
|}


Some numbers, like 36, can be arranged both as a square and as a triangle (see [[square triangular number]]):
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


:{|
==Demos==
|- align="center" valign="bottom"
|[[Image:GrayDot.svg|16px|*]][[Image:GrayDot.svg|16px|*]][[Image:GrayDot.svg|16px|*]][[Image:GrayDot.svg|16px|*]][[Image:GrayDot.svg|16px|*]][[Image:GrayDot.svg|16px|*]]<br>[[Image:GrayDot.svg|16px|*]][[Image:GrayDot.svg|16px|*]][[Image:GrayDot.svg|16px|*]][[Image:GrayDot.svg|16px|*]][[Image:GrayDot.svg|16px|*]][[Image:GrayDot.svg|16px|*]]<br>[[Image:GrayDot.svg|16px|*]][[Image:GrayDot.svg|16px|*]][[Image:GrayDot.svg|16px|*]][[Image:GrayDot.svg|16px|*]][[Image:GrayDot.svg|16px|*]][[Image:GrayDot.svg|16px|*]]<br>[[Image:GrayDot.svg|16px|*]][[Image:GrayDot.svg|16px|*]][[Image:GrayDot.svg|16px|*]][[Image:GrayDot.svg|16px|*]][[Image:GrayDot.svg|16px|*]][[Image:GrayDot.svg|16px|*]]<br>[[Image:GrayDot.svg|16px|*]][[Image:GrayDot.svg|16px|*]][[Image:GrayDot.svg|16px|*]][[Image:GrayDot.svg|16px|*]][[Image:GrayDot.svg|16px|*]][[Image:GrayDot.svg|16px|*]]<br>[[Image:GrayDot.svg|16px|*]][[Image:GrayDot.svg|16px|*]][[Image:GrayDot.svg|16px|*]][[Image:GrayDot.svg|16px|*]][[Image:GrayDot.svg|16px|*]][[Image:GrayDot.svg|16px|*]]
|
|[[Image:GrayDotX.svg|16px|*]]<br>[[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]]<br>[[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]]<br>[[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]]<br>[[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]]<br>[[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]]<br>[[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]]<br>[[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]]
|}


By convention, 1 is the first polygonal number for any number of sides. The rule for enlarging the polygon to the next size is to extend two adjacent arms by one point and to then add the required extra sides between those points. In the following diagrams, each extra layer is shown as in red.
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


===Triangular numbers===
[[File:Polygonal Number 3.gif|500px|none]]
<br>


===Square numbers===
* accessibility:
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


[[File:Polygonal Number 4.gif|500px|none]]
==Test pages ==
<br>


Polygons with higher numbers of sides, such as pentagons and hexagons, can also be constructed according to this rule, although the dots will no longer form a perfectly regular lattice like above.
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


===Pentagonal numbers===
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
[[File:Polygonal Number 5.gif|500px|none]]
==Bug reporting==
<br>
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
<br>
<br>
<br>
<br>
 
===Hexagonal numbers===
 
[[File:Polygonal Number 6.gif|500px|none]]
<br>
 
==Formula==
 
If ''s'' is the number of sides in a polygon, the formula for the ''n''<sup>th</sup> ''s''-gonal number ''P''(''s'',''n'') is
 
:<math>P(s,n) = \frac{n^2(s-2)-n(s-4)}{2}</math>
 
or
:<math>P(s,n) = (s-2)\frac{n(n-1)}{2}+n</math>
 
The ''n''<sup>th</sup> ''s''-gonal number is also related to the triangular numbers ''T''<sub>''n''</sub> as follows:
 
:<math>P(s,n) = (s-2)T_{n-1} + n = (s-3)T_{n-1} + T_n\, .</math>
 
Thus:
 
:<math>P(s,n+1)-P(s,n) = (s-2)n + 1\, ,</math>
:<math>P(s+1,n) - P(s,n) = T_{n-1} = \frac{n(n-1)}{2}\, .</math>
 
For a given ''s''-gonal number ''P''(''s'',''n'') = ''x'', one can find ''n'' by
 
:<math>n = \frac{\sqrt{8(s-2)x+(s-4)^2}+(s-4)}{2(s-2)}.</math>
 
==Table of values==
{| class="wikitable" border="1"
|-
! s
! Name
! Formula
! align="right" | ''n'' = 1
! align="right" | ''n'' = 2
! align="right" | ''n'' = 3
! align="right" | ''n'' = 4
! align="right" | ''n'' = 5
! align="right" | ''n'' = 6
! align="right" | ''n'' = 7
! align="right" | ''n'' = 8
! align="right" | ''n'' = 9
! align="right" | ''n'' = 10
! align="right" | Sum of Reciprocals<ref>[http://www.math.psu.edu/sellersj/downey_ong_sellers_cmj_preprint.pdf Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers]</ref>
! align="center" | [[On-Line Encyclopedia of Integer Sequences|OEIS]] number
|-
| align="right" | 3
| [[Triangular number|Triangular]]
|  ½(''n''²+''n'')
| align="right" | 1
| align="right" | 3
| align="right" | 6
| align="right" | 10
| align="right" | 15
| align="right" | 21
| align="right" | 28
| align="right" | 36
| align="right" | 45
| align="right" | 55
! align="right" | <math>{2}</math>
| {{OEIS link|id=A000217}}
|-
| align="right" | 4
| [[Square number|Square]]
| ''n''² = ½(2''n''² - 0''n'')
| align="right" | 1
| align="right" | 4
| align="right" | 9
| align="right" | 16
| align="right" | 25
| align="right" | 36
| align="right" | 49
| align="right" | 64
| align="right" | 81
| align="right" | 100
! align="right" | <math>{\pi^2\over6}</math>
| {{OEIS link|id=A000290}}
|-
| align="right" | 5
| [[Pentagonal number|Pentagonal]]
| ½(3''n''² - ''n'')
| align="right" | 1
| align="right" | 5
| align="right" | 12
| align="right" | 22
| align="right" | 35
| align="right" | 51
| align="right" | 70
| align="right" | 92
| align="right" | 117
| align="right" | 145
! align="right" | <math>{ 3\ln\left(3\right)}-{\pi\sqrt{3}\over3 }</math>
| {{OEIS link|id=A000326}}
|-
| align="right" | 6
| [[Hexagonal number|Hexagonal]]
| ½(4''n''² - 2''n'')
| align="right" | 1
| align="right" | 6
| align="right" | 15
| align="right" | 28
| align="right" | 45
| align="right" | 66
| align="right" | 91
| align="right" | 120
| align="right" | 153
| align="right" | 190
! align="right" | <math>{ 2\ln\left(2\right) }</math>
| {{OEIS link|id=A000384}}
|-
| align="right" | 7
| [[Heptagonal number|Heptagonal]]
| ½(5''n''² - 3''n'')
| align="right" | 1
| align="right" | 7
| align="right" | 18
| align="right" | 34
| align="right" | 55
| align="right" | 81
| align="right" | 112
| align="right" | 148
| align="right" | 189
| align="right" | 235
! align="right" | <math>\begin{matrix}
\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) \\
+\frac{1}{15}{\pi}{\sqrt{25-10\sqrt{5}}}
\end{matrix}</math><ref>http://www.siam.org/journals/problems/downloadfiles/07-003s.pdf</ref>
| {{OEIS link|id=A000566}}
|-
| align="right" | 8
| [[Octagonal number|Octagonal]]
| ½(6''n''² - 4''n'')
| align="right" | 1
| align="right" | 8
| align="right" | 21
| align="right" | 40
| align="right" | 65
| align="right" | 96
| align="right" | 133
| align="right" | 176
| align="right" | 225
| align="right" | 280
! align="right" | <math>{ {3\ln\left(3\right)\over4} + {\pi\sqrt{3}\over12} }</math>
| {{OEIS link|id=A000567}}
|-
| align="right" | 9
| [[Nonagonal number|Nonagonal]]
| ½(7''n''² - 5''n'')
| align="right" | 1
| align="right" | 9
| align="right" | 24
| align="right" | 46
| align="right" | 75
| align="right" | 111
| align="right" | 154
| align="right" | 204
| align="right" | 261
| align="right" | 325
! align="right" |
| {{OEIS link|id=A001106}}
|-
| align="right" | 10
| [[Decagonal number|Decagonal]]
| ½(8''n''² - 6''n'')
| align="right" | 1
| align="right" | 10
| align="right" | 27
| align="right" | 52
| align="right" | 85
| align="right" | 126
| align="right" | 175
| align="right" | 232
| align="right" | 297
| align="right" | 370
! align="right" | <math>{ {\ln\left(2\right)} + {\pi\over6} }</math>
| {{OEIS link|id=A001107}}
|-
| align="right" | 11
| Hendecagonal
| ½(9''n''² - 7''n'')
| align="right" | 1
| align="right" | 11
| align="right" | 30
| align="right" | 58
| align="right" | 95
| align="right" | 141
| align="right" | 196
| align="right" | 260
| align="right" | 333
| align="right" | 415
! align="right" |
| {{OEIS link|id=A051682}}
|-
| align="right" | 12
| [[Dodecagonal number|Dodecagonal]]
| ½(10''n''² - 8''n'')
| align="right" | 1
| align="right" | 12
| align="right" | 33
| align="right" | 64
| align="right" | 105
| align="right" | 156
| align="right" | 217
| align="right" | 288
| align="right" | 369
| align="right" | 460
! align="right" | 
| {{OEIS link|id=A051624}}
|-
| align="right" | 13
| Tridecagonal
| ½(11''n''² - 9''n'')
| align="right" | 1
| align="right" | 13
| align="right" | 36
| align="right" | 70
| align="right" | 115
| align="right" | 171
| align="right" | 238
| align="right" | 316
| align="right" | 405
| align="right" | 505
! align="right" | 
| {{OEIS link|id=A051865}}
|-
| align="right" | 14
| Tetradecagonal
| ½(12''n''² - 10''n'')
| align="right" | 1
| align="right" | 14
| align="right" | 39
| align="right" | 76
| align="right" | 125
| align="right" | 186
| align="right" | 259
| align="right" | 344
| align="right" | 441
| align="right" | 550
! align="right" | <math>{ {2\ln\left(2\right)\over5} + {3\ln\left(3\right)\over 10} + {\pi\sqrt{3}\over10} }</math>
| {{OEIS link|id=A051866}}
|-
| align="right" | 15
| Pentadecagonal
| ½(13''n''² - 11''n'')
| align="right" | 1
| align="right" | 15
| align="right" | 42
| align="right" | 82
| align="right" | 135
| align="right" | 201
| align="right" | 280
| align="right" | 372
| align="right" | 477
| align="right" | 595
! align="right" |
| {{OEIS link|id=A051867}}
|-
| align="right" | 16
| Hexadecagonal
| ½(14''n''² - 12''n'')
| align="right" | 1
| align="right" | 16
| align="right" | 45
| align="right" | 88
| align="right" | 145
| align="right" | 216
| align="right" | 301
| align="right" | 400
| align="right" | 513
| align="right" | 640
! align="right" |
| {{OEIS link|id=A051868}}
|-
| align="right" | 17
| Heptadecagonal
| ½(15''n''² - 13''n'')
| align="right" | 1
| align="right" | 17
| align="right" | 48
| align="right" | 94
| align="right" | 155
| align="right" | 231
| align="right" | 322
| align="right" | 428
| align="right" | 549
| align="right" | 685
! align="right" |
| {{OEIS link|id=A051869}}
|-
| align="right" | 18
| Octadecagonal
| ½(16''n''² - 14''n'')
| align="right" | 1
| align="right" | 18
| align="right" | 51
| align="right" | 100
| align="right" | 165
| align="right" | 246
| align="right" | 343
| align="right" | 456
| align="right" | 585
| align="right" | 730
! align="right" | <math>\begin{matrix}
\frac{4}{7}\ln{\left(2\right)} \\
-\frac{\sqrt{2}}{14}\ln\left(3 - 2\sqrt{2}\right) \\
+\pi \frac{\left(1 + \sqrt{2}\right)}{14}
\end{matrix} </math>
| {{OEIS link|id=A051870}}
|-
| align="right" | 19
| Nonadecagonal
| ½(17''n''² - 15''n'')
| align="right" | 1
| align="right" | 19
| align="right" | 54
| align="right" | 106
| align="right" | 175
| align="right" | 261
| align="right" | 364
| align="right" | 484
| align="right" | 621
| align="right" | 775
! align="right" |
| {{OEIS link|id=A051871}}
|-
| align="right" | 20
| Icosagonal
| ½(18''n''² - 16''n'')
| align="right" | 1
| align="right" | 20
| align="right" | 57
| align="right" | 112
| align="right" | 185
| align="right" | 276
| align="right" | 385
| align="right" | 512
| align="right" | 657
| align="right" | 820
! align="right" |
| {{OEIS link|id=A051872}}
|-
| align="right" | 21
| Icosihenagonal
| ½(19''n''² - 17''n'')
| align="right" | 1
| align="right" | 21
| align="right" | 60
| align="right" | 118
| align="right" | 195
| align="right" | 291
| align="right" | 406
| align="right" | 540
| align="right" | 693
| align="right" | 865
! align="right" |
| {{OEIS link|id=A051873}}
|-
| align="right" | 22
| Icosidigonal
| ½(20''n''² - 18''n'')
| align="right" | 1
| align="right" | 22
| align="right" | 63
| align="right" | 124
| align="right" | 205
| align="right" | 306
| align="right" | 427
| align="right" | 568
| align="right" | 729
| align="right" | 910
! align="right" |
| {{OEIS link|id=A051874}}
|-
| align="right" | 23
| Icositrigonal
| ½(21''n''² - 19''n'')
| align="right" | 1
| align="right" | 23
| align="right" | 66
| align="right" | 130
| align="right" | 215
| align="right" | 321
| align="right" | 448
| align="right" | 596
| align="right" | 765
| align="right" | 955
! align="right" |
| {{OEIS link|id=A051875}}
|-
| align="right" | 24
| Icositetragonal
| ½(22''n''² - 20''n'')
| align="right" | 1
| align="right" | 24
| align="right" | 69
| align="right" | 136
| align="right" | 225
| align="right" | 336
| align="right" | 469
| align="right" | 624
| align="right" | 801
| align="right" | 1000
! align="right" |
| {{OEIS link|id=A051876}}
|-
| align="right" | 10000
| Myriagonal
| ½(9998''n''² - 9996''n'')
| align="right" | 1
| align="right" | 10000
| align="right" | 29997
| align="right" | 59992
| align="right" | 99985
| align="right" | 149976
| align="right" | 209965
| align="right" | 279952
| align="right" | 359937
| align="right" | 449920
! align="right" |
| {{OEIS link|id=A167149}}
|}
 
The [[On-Line Encyclopedia of Integer Sequences]] eschews terms using Greek prefixes (e.g., "octagonal") in favor of terms using numerals (i.e., "8-gonal").
 
 
A property of this table can be expressed by the following identity (see {{OEIS link|id=A086270}}):
 
:<math>2\,P(s,n) = P(s+k,n) + P(s-k,n),</math>
 
with
 
:<math>k = 0, 1, 2, 3, ..., s-3.</math>
 
==Combinations==
Some numbers, such as 36 which is both square and triangular, fall into two polygonal sets. The problem of determining, given two such sets, all numbers that belong to both can be solved by reducing the problem to [[Pell's equation]]. The simplest example of this is the sequence of [[square triangular number]]s.
 
The following table summarizes the set of ''s''-gonal ''t''-gonal numbers for small values of ''s'' and ''t''.
{| class="wikitable" border="1"
|-
! ''s''
! ''t''
! Sequence
! [[On-Line Encyclopedia of Integer Sequences|OEIS]] number
|-
| 4
| 3
| 1, 36, 1225, 41616, 1413721, 48024900, 1631432881 , 55420693056, 1882672131025, 63955431761796, 2172602007770041, 73804512832419600, 2507180834294496361, 85170343853180456676, 2893284510173841030625, 98286503002057414584576, 3338847817559778254844961, ...
| {{OEIS link|id=A001110}}
|-
| 5
| 3
| 1, 210, 40755, 7906276, …
| {{OEIS link|id=A014979}}
|-
| 5
| 4
| 1, 9801, 94109401, 903638458801, 8676736387298001, 83314021887196947001, 799981229484128697805801, ...
| {{OEIS link|id=A036353}}
|-
| 6
| 3
| All hexagonal numbers are also triangular.
| {{OEIS link|id=A000384}}
|-
| 6
| 4
| Odd triangular square numbers.
| {{OEIS link|id=A046177}}
|-
| 6
| 5
| 1, 40755, 1533776805, …
| {{OEIS link|id=A046180}}
|-
| 7
| 3
| 1, 55, 121771, 5720653, …
| {{OEIS link|id=A046194}}
|-
| 7
| 4
| 1, 81, 5929, 2307361, 168662169, 12328771225, 4797839017609, 350709705290025, 25635978392186449, 9976444135331412025, 729252434211108535809, 53306479301521270428241, 20744638830126197732344369, 1516379800105728357531817761, 110843467413344235941816109721, 43135613687078894324987720634481, 3153102533906718276539864534846601, …
| {{OEIS link|id=A036354}}
|-
| 7
| 5
| 1, 4347, 16701685, 64167869935, …
| {{OEIS link|id=A048900}}
|-
| 7
| 6
| 1, 121771, 12625478965, …
| {{OEIS link|id=A048903}}
|-
| 8
| 3
| 1, 21, 11781, 203841, …
| {{OEIS link|id=A046183}}
|-
| 8
| 4
| 1, 225, 43681, 8473921, 1643897025, 318907548961, 61866420601441, 12001766689130625, 2328280871270739841, 451674487259834398561, 87622522247536602581025, 16998317641534841066320321,  …
| {{OEIS link|id=A036428}}
|-
| 8
| 5
| 1, 176, 1575425, 234631320, …
| {{OEIS link|id=A046189}}
|-
| 8
| 6
| 1, 11781, 113123361, …
| {{OEIS link|id=A046192}}
|-
| 8
| 7
| 1, 297045, 69010153345, …
| {{OEIS link|id=A048906}}
|-
| 9
| 3
| 1, 325, 82621, 20985481, …
| {{OEIS link|id=A048909}}
|-
| 9
| 4
| 1, 9, 1089, 8281, 978121, 7436529, 878351769, 6677994961, 788758910641, 5996832038649, 708304623404049, 5385148492712041, 636056763057925561, 4835857349623374369, 571178264921393749929, 4342594514813297471521, 512917445842648529510881, 3899645038444991506051689, 460599295188433458107021409, 3501876901929087559136945401,  …
| {{OEIS link|id=A036411}}
|-
| 9
| 5
| 1, 651, 180868051, …
| {{OEIS link|id=A048915}}
|-
| 9
| 6
| 1, 325, 5330229625, …
| {{OEIS link|id=A048918}}
|-
| 9
| 7
| 1, 26884, 542041975, …
| {{OEIS link|id=A048921}}
|-
| 9
| 8
| 1, 631125, 286703855361, …
| {{OEIS link|id=A048924}}
|-
|10
|3
| 1, 10, 1540, 1777555, 13773376, 2051297326, 15894464365, 2367195337045, 18342198104230, ...
|-
|10
|4
| One and no other.
|-
| 11
| 4
| 1, 196 , 29241, 1755625, ...
|-
| 12
| 4
| 1, 64, 3025 , 142129, 6677056, 313679521, 14736260449, 692290561600, 32522920134769, 1527884955772561, 71778070001175616, 3372041405099481409, 158414167969674450625, 7442093853169599697984, 349619996931001511354641, 16424697761903901433970161, 771611174812552365885242944, 36249300518428057295172448225, 1702945513191306140507219823649, 80002189819472960546544159263296, 3758399976002037839547068265551281, 176564796682276305498165664321646929, 8294787044090984320574239154851854400, 389678426275593986761491074613715509889, 18306591247908826393469506267689777110401, 860020110225439246506305303506805808678976, 40402638589347735759402879758552183230801489, 1898063993589118141445429043348445806038991025, 89168605060099204912175762157618400700601776704, ...
|-
| 13
| 4
| 1, 36, 35721, 34999056, 896703025, 34291262041, ...
|-
| 14
| 4
| 1, 441, 14161, 4239481, 135978921, 40707501121, 1305669590281, 390873421529361, 12537039269904241, 3753166552817428201, 1155894986496417625263961, ...
|-
| 15
| 4
| 1 , 3025 , 5997601, 165148201, ...
|-
|16
|4
| 1, 16, 400, 4225, 101761, ...
|-
|18
|4
| 1, 100, 1936, 116281, 2235025, 134189056, 2579217796, 154854055225, 2976415102441, 178701445541476, 3434780449000000, ...
|-
|22
|4
| 1, 729, 284089, 3900625, 15175959521, 590725976569, 8110813506601, 3156387347610225, 1228333148092290241, 16865317394711073289, 6563271907899976822281, 2554149271482890096235025, 35069100108493095964960369, ...
|-
|28
|4
|1, 81, 3136, 30625, ...
|-
|30
|4
| 1, 203401, 1819801, 164024190001, 1467492382801, 132269434866199801, 1183388792474889001, 106662336814809228952801, 954287089027867949018401, 86012721732003522411131649001, 769539017165067381031862931001, 69360830830024442142566574789968401, 620557802518990379109828463337266801, 55932712702907357470917967521368968071001, 500419053066149340677758825111066761145801, ...
|-
|32
|4
| 1, 1089, 9025, 4190209, 34680321, 16098788161, 133241790529, 61851539930625, 511914924538369 , 237633600314679361, 1966777006834629441, 912988230557458180609, 7556356748343721780225, 3507700544168154015226689, 29031520660359572245001281, 13476584577705817169042764801, 111539094820744728221573147649, 51777034439845205395308287145025, 428533173269780585467711788272449, 198927352841300701422957270168427521, 1646424340163402188622220468969607681, 764278837839242855021796436678811396929, 6325561886374617938905985574069444444225, 2936359096051018207693040486762723218579969, ...
|-
|40
|4
|1, 576, 123201, ...
|-
|44
|4
| 1, 256, 1521, 136161, 802816, 71757841, 423083761, 37816247296, 222964340481, ...
|-
|50
|4
| 1, 5776, 30276, 55487601, 290736601, 532791965476, 2791652838976, 5115868397039401, 26805450269137401, 49122567815580389376, 257385930692604511876, 471674891049334501775401, 2471419679704938253922401, 4529022254733142070467037476, 23730571507140886421558408976, 43487671218272739111289992095601, 227860945140147111714865589091601, ...
|-
|64
|4
| 1, 64, 625, 48400, 450241, ...
|-
| 66
| 4
| 1, 1223236, 5107600, 1629005505625, 6801867425521, 2169369437921667136, 9058142076710164516, 2888979651650786027844601, ...
|-
|68
|4
| 1, 400, 41616, 4289041, 17514225, ...
|-
| 96
| 4
| 1, 14400, 46656, 132733441, 429940225, ...
|-
|128
|4
|1, 148225, 408321, 9563079681, 26342913025, 616952522883841, 1699486690978561, 39802075051765530625, 109640684355448463361, 2567791069272648920349441, 7073359108807915474785025, 165658473003253597395658798081, 456330689435993174584833131521, 10687290724764111513110882779540225, 29439718091200304556358009172652801, 689479873651773417153581894243599769601, 1899273972479365758712887429179690164225,  ...
|-
|132
|4
| 1, 784, 262144, 10597261249, 28731945025, ...
|-
|140
|4
|1, 1002001, 2637376, 1023640086001, ...
|-
|156
|4
| 1, 18496, 288456256, ...
 
|}
In some cases, such as ''s''=10 and ''t''=4, there are no numbers in both sets other than 1.
 
The problem of finding numbers that belong to three polygonal sets is more difficult. A computer search for pentagonal square triangular numbers has yielded only the trivial value of 1, though a proof that there are no other such numbers has yet to appear in print.<ref>{{MathWorld|title=Pentagonal Square Triangular Number | urlname=PentagonalSquareTriangularNumber}}</ref>
 
The number 1225 is hecticositetragonal (''s''=124), hexacontagonal (''s''=60), icosienneagonal (''s''=29), hexagonal, square, and triangular.
 
==See also==
 
* [[Polyhedral number]]
* [[Fermat polygonal number theorem]]
 
==Notes==
{{reflist}}
 
==References==
*''[[The Penguin Dictionary of Curious and Interesting Numbers]]'', David Wells ([[Penguin Books]], 1997) [ISBN 0-14-026149-4].
*[http://planetmath.org/encyclopedia/PolygonalNumber.html Polygonal numbers at PlanetMath]
*{{MathWorld | title=Polygonal Numbers | urlname=PolygonalNumber}}
*{{cite book|author=F. Tapson|title=The Oxford Mathematics Study Dictionary|publisher=Oxford University Press|year=1999|page=88-89|edition=2nd|isbn=0-19-914-567-9}}
 
==External links==
* {{springer|title=Polygonal number|id=p/p073600}}
*[http://www.virtuescience.com/polygonal-numbers.html Polygonal Numbers: Every s-polygonal number between 1 and 1000 clickable for 2<=s<=337]
*{{youtube|id=YOiZ459lZ7A|title=Polygonal Numbers on the Ulam Spiral grid}}
* Polygonal Number Counting Function: http://www.mathisfunforum.com/viewtopic.php?id=17853
{{Classes of natural numbers}}
[[Category:Figurate numbers]]
[[Category:Recreational mathematics]]

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