View source for Polygonal number

{{Short description|Type of figurate number}}
In [[mathematics]], a '''polygonal number''' is a [[Integer|number]] that counts dots arranged in the shape of a [[regular polygon]]{{r|tattersall2005|p=2-3}}. These are one type of 2-dimensional [[figurate number]]s.

Polygonal numbers were first studied during the 6th century BC by the Ancient Greeks, who investigated and discussed properties of [[Pronic number|oblong]], [[Triangular Number|triangular]], and [[Square number|square numbers]]{{r|tattersall2005|p=1}}.

== Definition and examples ==

The number 10 for example, can be arranged as a [[triangle]] (see [[triangular number]]):

:{|
| align="center" style="line-height: 0;" | [[File:GrayDotX.svg|16px|*]]<br>[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]]<br>[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]]<br>[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]]
|}

But 10 cannot be arranged as a [[square (geometry)|square]]. The number 9, on the other hand, can be (see [[square number]]):

:{|
| align="center" style="line-height: 0;" | [[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]]<br>[[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]]<br>[[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]]
|}

Some numbers, like 36, can be arranged both as a square and as a triangle (see [[square triangular number]]):

:{| cellpadding="5"
|- align="center" valign="bottom"
| style="line-height: 0; display: inline-block;"|[[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]]<br>[[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]]<br>[[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]]<br>[[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]]<br>[[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]]<br>[[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]]
| style="line-height: 0; display: inline-block"|[[File:GrayDotX.svg|16px|*]]<br>[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]]<br>[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]]<br>[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]]<br>[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]]<br>[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]]<br>[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]]<br>[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File: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.

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

The triangular number sequence is the representation of the numbers in the form of equilateral triangle arranged in a series or sequence. These numbers are in a sequence of 1, 3, 6, 10, 15, 21, 28, 36, 45, and so on.

===Square numbers===

:[[File:Polygonal Number 4.gif|500px|none]]

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.

===Pentagonal numbers===

:[[File:Polygonal Number 5.gif|500px|none]]

===Hexagonal numbers===

:[[File:Polygonal Number 6.gif|500px|none]]

==Formula==
[[File:visual_proof_polygonal_numbers.svg|thumb|An ''s''-gonal number greater than 1 can be decomposed into ''s''&minus;2 triangular numbers and a natural number.]]If {{mvar|s}} is the number of sides in a polygon, the formula for the {{mvar|n}}th {{mvar|s}}-gonal number {{math|''P''(''s'',''n'')}} is

:<math>P(s,n) = \frac{(s-2)n^2-(s-4)n}{2}</math>

The {{mvar|n}}th {{mvar|s}}-gonal number is also related to the triangular numbers {{math|''T''<sub>''n''</sub>}} as follows:<ref name=":0">{{Cite book |last1=Conway |first1=John H. |title=The Book of Numbers |last2=Guy |first2=Richard |date=2012-12-06 |publisher=Springer Science & Business Media |isbn=978-1-4612-4072-3 |pages=38–41 |language=en |author-link=John Horton Conway |author-link2=Richard K. Guy}}</ref>

:<math>P(s,n) = (s-2)T_{n-1} + n = (s-3)T_{n-1} + T_n\, .</math>

Thus:

:<math>\begin{align}
P(s,n+1)-P(s,n) &= (s-2)n + 1\, ,\\
P(s+1,n) - P(s,n) &= T_{n-1} = \frac{n(n-1)}{2}\, ,\\
P(s+k,n) - P(s,n) &= k T_{n-1} = k\frac{n(n-1)}{2}\, .
\end{align}</math>

For a given {{mvar|s}}-gonal number {{math|''P''(''s'',''n'') {{=}} ''x''}}, one can find {{mvar|n}} by

:<math>n = \frac{\sqrt{8(s-2)x+{(s-4)}^2}+(s-4)}{2(s-2)}</math>

and one can find {{mvar|s}} by

:<math>s = 2+\frac{2}{n}\cdot\frac{x-n}{n-1}</math>.

===Every hexagonal number is also a triangular number===
{{CSS image crop
|Image   = hexagonal_number_visual_proof.svg
|bSize   = 340
|cWidth  = 200
|cHeight = 300
|oTop    =   0
|oLeft   = 128
|Description = [[Proof without words]] that hexagonal numbers are odd-sided triangular numbers
}}
Applying the formula above:
:<math>P(s,n) = (s-2)T_{n-1} + n </math>

to the case of 6 sides gives:
:<math>P(6,n) = 4T_{n-1} + n </math>

but since:
:<math>T_{n-1} = \frac{n(n-1)}{2} </math>

it follows that:
:<math>P(6,n) = \frac{4n(n-1)}{2} + n = \frac{2n(2n-1)}{2} = T_{2n-1}</math>

This shows that the {{mvar|n}}th hexagonal number {{math|''P''(6,''n'')}} is also the {{math|(2''n'' − 1)}}th triangular number {{math|''T''<sub>2''n''−1</sub>}}.  We can find every hexagonal number by simply taking the odd-numbered triangular numbers:<ref name=":0" />

:'''1''', 3, '''6''', 10, '''15''', 21, '''28''', 36, '''45''', 55, '''66''', ...

==Table of values==
The first 6 values in the column "sum of reciprocals", for triangular to octagonal numbers, come from a published solution to the general problem, which also gives a general formula for any number of sides, in terms of the [[digamma function]].<ref name="siam_07-003s">{{Cite web |url=http://www.siam.org/journals/problems/downloadfiles/07-003s.pdf |title=Sums of Reciprocals of Polygonal Numbers and a Theorem of Gauss |access-date=2010-06-13 |archive-url=https://web.archive.org/web/20110615085610/http://www.siam.org/journals/problems/downloadfiles/07-003s.pdf |archive-date=2011-06-15 |url-status=dead }}</ref>
{| class="wikitable"
|-
! rowspan="2"|{{mvar|s}}
! rowspan="2"|Name
! rowspan="2"|Formula
! colspan="10"| {{mvar|n}} 
! rowspan="2" align="right" | Sum of reciprocals<ref name="siam_07-003s" /><ref>{{Cite web |url=http://www.math.psu.edu/sellersj/downey_ong_sellers_cmj_preprint.pdf |title=Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers |access-date=2010-05-13 |archive-date=2013-05-29 |archive-url=https://web.archive.org/web/20130529032918/http://www.math.psu.edu/sellersj/downey_ong_sellers_cmj_preprint.pdf |url-status=dead }}</ref>
! rowspan="2" align="center" | [[On-Line Encyclopedia of Integer Sequences|OEIS]] number
|- 
! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !! 10
|-
| align="right" | [[Digon|2]]
| [[Natural number|Natural]] (line segment)
| align="center" | {{math|{{sfrac|1|2}}(''0n''<sup>2</sup> + ''2n'') {{=}} ''n''}}
| align="right" | 1
| align="right" | 2
| align="right" | 3
| align="right" | 4
| align="right" | 5
| align="right" | 6
| align="right" | 7
| align="right" | 8
| align="right" | 9
| align="right" | 10
| align="center" | ∞ ([[Harmonic series (mathematics)|diverges]])
| {{OEIS link|id=A000027}}
|-
| align="right" | [[triangle|3]]
| [[Triangular number|Triangular]]
| align="center" | {{math|{{sfrac|1|2}}(''n''<sup>2</sup> + ''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="center" | 2<ref name="siam_07-003s" />
| {{OEIS link|id=A000217}}
|-
| align="right" | [[quadrilateral|4]]
| [[Square number|Square]]
| align="center" | {{math|{{sfrac|1|2}}(2''n''<sup>2</sup> − 0''n'') <br> {{=}} ''n''<sup>2</sup>}}
| 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="center" | {{sfrac|{{pi}}<sup>2</sup>|6}}<ref name="siam_07-003s" />
| {{OEIS link|id=A000290}}
|-
| align="right" | [[pentagon|5]]
| [[Pentagonal number|Pentagonal]]
| align="center" | {{math|{{sfrac|1|2}}(3''n''<sup>2</sup> − ''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="center" | {{math|3 [[natural logarithm|ln]] 3 − {{sfrac|{{pi}}{{sqrt|3}}|3}}}}<ref name="siam_07-003s" />
| {{OEIS link|id=A000326}}
|-
| align="right" | [[hexagon|6]]
| [[Hexagonal number|Hexagonal]]
| align="center" | {{math|{{sfrac|1|2}}(4''n''<sup>2</sup> − 2''n'') <br> {{=}} 2''n''<sup>2</sup> - ''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="center" | {{math|2 ln 2}}<ref name="siam_07-003s" />
| {{OEIS link|id=A000384}}
|-
| align="right" | [[heptagon|7]]
| [[Heptagonal number|Heptagonal]]
| align="center" | {{math|{{sfrac|1|2}}(5''n''<sup>2</sup> − 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="center" | <math>\begin{matrix}
\tfrac{2}{3}\ln 5  \\ 
+\tfrac{{1}+\sqrt{5}}{3}\ln\tfrac\sqrt{10-2\sqrt{5}}{2} \\ 
+\tfrac{{1}-\sqrt{5}}{3}\ln\tfrac\sqrt{10+2\sqrt{5}}{2} \\
+\tfrac{\pi\sqrt{25-10\sqrt{5}}}{15}
\end{matrix}</math><ref name="siam_07-003s" />
| {{OEIS link|id=A000566}}
|-
| align="right" | [[octagon|8]]
| [[Octagonal number|Octagonal]]
| align="center" | {{math|{{sfrac|1|2}}(6''n''<sup>2</sup> − 4''n'') <br> {{=}} 3''n''<sup>2</sup> - 2''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="center" | {{math|{{sfrac|3|4}} ln 3 + {{sfrac|{{pi}}{{sqrt|3}}|12}}}}<ref name="siam_07-003s" />
| {{OEIS link|id=A000567}}
|-
| align="right" | [[nonagon|9]]
| [[Nonagonal number|Nonagonal]]
| align="center" | {{math|{{sfrac|1|2}}(7''n''<sup>2</sup> − 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="center" | 
| {{OEIS link|id=A001106}}
|-
| align="right" | [[decagon|10]]
| [[Decagonal number|Decagonal]]
| align="center" | {{math|{{sfrac|1|2}}(8''n''<sup>2</sup> − 6''n'') <br> {{=}} 4''n''<sup>2</sup> - 3''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="center" | {{math|ln 2 + {{sfrac|{{pi}}|6}}}}
| {{OEIS link|id=A001107}} 
|-
| align="right" | [[hendecagon|11]]
| Hendecagonal
| align="center" | {{math|{{sfrac|1|2}}(9''n''<sup>2</sup> − 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="center" |
| {{OEIS link|id=A051682}}
|-
| align="right" | [[dodecagon|12]]
| [[Dodecagonal number|Dodecagonal]]
| align="center" | {{math|{{sfrac|1|2}}(10''n''<sup>2</sup> − 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="center" |  
| {{OEIS link|id=A051624}}
|-
| align="right" | [[tridecagon|13]]
| Tridecagonal
| align="center" | {{math|{{sfrac|1|2}}(11''n''<sup>2</sup> − 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="center" |  
| {{OEIS link|id=A051865}}
|-
| align="right" | [[tetradecagon|14]]
| Tetradecagonal
| align="center" | {{math|{{sfrac|1|2}}(12''n''<sup>2</sup> − 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="center" | {{math|{{sfrac|2|5}} ln 2 + {{sfrac|3|10}} ln 3  + {{sfrac|{{pi}}{{sqrt|3}}|10}}}}
| {{OEIS link|id=A051866}}
|-
| align="right" | [[pentadecagon|15]]
| Pentadecagonal
| align="center" | {{math|{{sfrac|1|2}}(13''n''<sup>2</sup> − 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="center" | 
| {{OEIS link|id=A051867}}
|-
| align="right" | [[hexadecagon|16]]
| Hexadecagonal
| align="center" | {{math|{{sfrac|1|2}}(14''n''<sup>2</sup> − 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="center" | 
| {{OEIS link|id=A051868}}
|-
| align="right" | [[heptadecagon|17]]
| Heptadecagonal
| align="center" | {{math|{{sfrac|1|2}}(15''n''<sup>2</sup> − 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="center" | 
| {{OEIS link|id=A051869}}
|-
| align="right" | [[octadecagon|18]]
| Octadecagonal
| align="center" | {{math|{{sfrac|1|2}}(16''n''<sup>2</sup> − 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="center" | {{math|{{sfrac|4|7}} ln 2 − {{sfrac|{{sqrt|2}}|14}} ln (3 − 2{{sqrt|2}})}} {{math|+ {{sfrac|{{pi}}(1 + {{sqrt|2}})|14}}}}
| {{OEIS link|id=A051870}}
|-
| align="right" | [[enneadecagon|19]]
| Enneadecagonal
| align="center" | {{math|{{sfrac|1|2}}(17''n''<sup>2</sup> − 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="center" | 
| {{OEIS link|id=A051871}}
|-
| align="right" | [[icosagon|20]]
| Icosagonal
| align="center" | {{math|{{sfrac|1|2}}(18''n''<sup>2</sup> − 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="center" | 
| {{OEIS link|id=A051872}}
|-
| align="right" | [[icosihenagon|21]]
| Icosihenagonal
| align="center" | {{math|{{sfrac|1|2}}(19''n''<sup>2</sup> − 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="center" | 
| {{OEIS link|id=A051873}}
|-
| align="right" | [[icosidigon|22]]
| Icosidigonal
| align="center" | {{math|{{sfrac|1|2}}(20''n''<sup>2</sup> − 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="center" | 
| {{OEIS link|id=A051874}}
|-
| align="right" | [[icositrigon|23]]
| Icositrigonal
| align="center" | {{math|{{sfrac|1|2}}(21''n''<sup>2</sup> − 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="center" | 
| {{OEIS link|id=A051875}}
|-
| align="right" | [[icositetragon|24]]
| Icositetragonal
| align="center" | {{math|{{sfrac|1|2}}(22''n''<sup>2</sup> − 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="center" | 
| {{OEIS link|id=A051876}}
|-
| align="right" | ...
| ...
| align="center" | ...
| align="right" | ...
| align="right" | ...
| align="right" | ...
| align="right" | ...
| align="right" | ...
| align="right" | ...
| align="right" | ...
| align="right" | ...
| align="right" | ...
| align="right" | ...
| align="center" | ...
| ...
|-
| align="right" | [[myriagon|10000]]
| Myriagonal
| align="center" | {{math|{{sfrac|1|2}}(9998''n''<sup>2</sup> − 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="center" | 
| {{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 {{mvar|s}}-gonal {{mvar|t}}-gonal numbers for small values of {{mvar|s}} and {{mvar|t}}.
:{| class="wikitable" border="1"
|-
! {{mvar|s}}
! {{mvar|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, 1533776805, 297544793910, 57722156241751, 11197800766105800, 2172315626468283465, …
| {{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
|1, 1225, 1413721, 1631432881, 1882672131025, 2172602007770041, 2507180834294496361, 2893284510173841030625, 3338847817559778254844961, 3853027488179473932250054441, ...
| {{OEIS link|id=A046177}}
|-
|6
|5
|1, 40755, 1533776805, …
| {{OEIS link|id=A046180}}
|-
|7
|3
|1, 55, 121771, 5720653, 12625478965, 593128762435, 1309034909945503, 61496776341083161, 135723357520344181225, 6376108764003055554511, 14072069153115290487843091, …
| {{OEIS link|id=A046194}}
|-
|7
|4
|1, 81, 5929, 2307361, 168662169, 12328771225, 4797839017609, 350709705290025, 25635978392186449, 9976444135331412025, …
| {{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, ...
| {{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}} 
|-
|}

In some cases, such as {{math|''s'' {{=}} 10}} and {{math|''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 be found.<ref>{{MathWorld|title=Pentagonal Square Triangular Number | urlname=PentagonalSquareTriangularNumber}}</ref>

The number [[1000_(number)#1200_to_1299|1225]] is hecatonicositetragonal ({{math|''s'' {{=}} 124}}), hexacontagonal ({{math|''s'' {{=}} 60}}), icosienneagonal ({{math|''s'' {{=}} 29}}), hexagonal, square, and triangular.

==See also==

* [[Centered polygonal number]]
*[[Polyhedral number]]
* [[Fermat polygonal number theorem]]
*

==Notes==
{{Reflist|refs=
<ref name="tattersall2005">
{{cite book
  | edition = 2nd
  | first = James J.
  | isbn = 978-0-511-75634-4
  | last = Tattersall 
  | location = New York
  | publisher = Cambridge University Press
  | title = Elementary Number Theory in Nine Chapters
  | year = 2005
}}
</ref>}}

==References==
*''[[The Penguin Dictionary of Curious and Interesting Numbers]]'', [[David G. Wells|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|pages=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}}
{{series (mathematics)}}
{{Authority control}}

[[Category:Figurate numbers]]