Editing Polygonal number (section)

==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>