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:<math>\frac{\pi^2}{8}</math>, if ''k'' = 8
:<math>\frac{\pi^2}{8}</math>, if ''k'' = 8

== Table of formulae and values ==

Centered polygonal numbers associated with [[constructible polygons]] (Cf. A003401) (with straightedge and compass) are named in '''bold'''.

<!--== Wikitable color scheme is: [background: #f2f2f2; for table header] [background: #f9f9f9; for table cells] ==-->
{| class="wikitable" align="center" border="1" cellspacing="0" cellpadding="4" style="border-collapse: collapse; border: 1px solid darkgray; background: #f9f9f9; color: black; empty-cells: show; text-align: right;"
|+ '''Centered polygonal numbers formulae and values'''
|- style="background: #f2f2f2; color: black; text-align: center;"
! width="25" style="text-align: center;" | ''N''<sub>0</sub>
! style="text-align: center;" | Name
! style="text-align: center;" | Formulae

<math>\,_cP^{(2)}_{N_0}(n)</math>
! width="50" align="center" | ''n'' = 0
! width="50" align="center" | 1
! width="50" align="center" | 2
! width="50" align="center" | 3
! width="50" align="center" | 4
! width="50" align="center" | 5
! width="50" align="center" | 6
! width="50" align="center" | 7
! width="50" align="center" | 8
! width="50" align="center" | 9
! width="50" align="center" | 10
! width="50" align="center" | 11
! width="75" style="text-align: center;" | OEIS
number
|-
| align="center" | '''3'''
| align="left" | [[Centered triangular numbers|'''Centered triangular''']]
| align="center" | <math>3T_n+1\,</math>
<math>3n(n+1)/2+1\,</math>
| align="right" | 1
| align="right" | 4
| align="right" | 10
| align="right" | 19
| align="right" | 31
| align="right" | 46
| align="right" | 64
| align="right" | 85
| align="right" | 109
| align="right" | 136
| align="right" | 166
| align="right" | 199
| align="center" | A005448('''''n'''''+1)
|-
| align="center" | '''4'''
| align="left" | [[Centered square numbers|'''Centered square''']]
| align="center" | <math>4T_n+1\,</math>
<math>2n(n+1)+1\,</math>

<math>n^2+(n+1)^2\,</math>
| align="right" | 1
| align="right" | 5
| align="right" | 13
| align="right" | 25
| align="right" | 41
| align="right" | 61
| align="right" | 85
| align="right" | 113
| align="right" | 145
| align="right" | 181
| align="right" | 221
| align="right" | 265
| align="center" | A001844('''''n''''')
|-
| align="center" | '''5'''
| align="left" | [[Centered pentagonal numbers|'''Centered pentagonal''']]
| align="center" | <math>5T_n+1\,</math>
<math>5n(n+1)/2+1\,</math>
| align="right" | 1
| align="right" | 6
| align="right" | 16
| align="right" | 31
| align="right" | 51
| align="right" | 76
| align="right" | 106
| align="right" | 141
| align="right" | 181
| align="right" | 226
| align="right" | 276
| align="right" | 331
| align="center" | A005891('''''n''''')
|-
| align="center" | '''6'''
| align="left" | [[Centered hexagonal numbers|'''Centered hexagonal''']]

[[Hex numbers|'''Hex numbers''']]
| align="center" | <math>6T_n+1\,</math>
<math>3n(n+1)+1\,</math>
| align="right" | 1
| align="right" | 7
| align="right" | 19
| align="right" | 37
| align="right" | 61
| align="right" | 91
| align="right" | 127
| align="right" | 169
| align="right" | 217
| align="right" | 271
| align="right" | 331
| align="right" | 397
| align="center" | A003215('''''n''''')
|-
| align="center" | 7
| align="left" | [[Centered heptagonal numbers|Centered heptagonal]]
| align="center" | <math>7T_n+1\,</math>
<math>7n(n+1)/2+1\,</math>
| align="right" | 1
| align="right" | 8
| align="right" | 22
| align="right" | 43
| align="right" | 71
| align="right" | 106
| align="right" | 148
| align="right" | 197
| align="right" | 253
| align="right" | 316
| align="right" | 386
| align="right" | 463
| align="center" | A069099('''''n'''''+1)
|-
| align="center" | '''8'''
| align="left" | [[Centered octagonal numbers|'''Centered octagonal''']]
| align="center" | <math>8T_n+1\,</math>
<math>4n(n+1)+1\,</math>

<math>(2n+1)^2\,</math>

[[Odd squares]]
| align="right" | 1
| align="right" | 9
| align="right" | 25
| align="right" | 49
| align="right" | 81
| align="right" | 121
| align="right" | 169
| align="right" | 225
| align="right" | 289
| align="right" | 361
| align="right" | 441
| align="right" | 529
| align="center" | A016754('''''n''''')
|-
| align="center" | 9
| align="left" | [[Centered nonagonal numbers|Centered nonagonal]]
| align="center" | <math>9T_n+1\,</math>
<math>9n(n+1)/2+1\,</math>

<math>t_{3n+1}\,</math>

<math>\binom{3n+2}{2}</math>
| align="right" | 1
| align="right" | 10
| align="right" | 28
| align="right" | 55
| align="right" | 91
| align="right" | 136
| align="right" | 190
| align="right" | 253
| align="right" | 325
| align="right" | 406
| align="right" | 496
| align="right" | 595
| align="center" | A060544('''''n'''''+1)
|-
| align="center" | '''10'''
| align="left" | '''[[Centered decagonal numbers|Centered decagonal]]'''
| align="center" | <math>10T_n+1\,</math>
<math>5n(n+1)+1\,</math>
| align="right" | 1
| align="right" | 11
| align="right" | 31
| align="right" | 61
| align="right" | 101
| align="right" | 151
| align="right" | 211
| align="right" | 281
| align="right" | 361
| align="right" | 451
| align="right" | 551
| align="right" | 661
| align="center" | A062786('''''n'''''+1)
|-
| align="center" | 11
| align="left" | [[Centered hendecagonal numbers|Centered hendecagonal]]
| align="center" | <math>11T_n+1\,</math>
<math>11n(n+1)/2+1\,</math>
| align="right" | 1
| align="right" | 12
| align="right" | 34
| align="right" | 67
| align="right" | 111
| align="right" | 166
| align="right" | 232
| align="right" | 309
| align="right" | 397
| align="right" | 496
| align="right" | 606
| align="right" | 727
| align="center" | A069125('''''n'''''+1)
|-
| align="center" | '''12'''
| align="left" | [[Centered dodecagonal numbers|'''Centered dodecagonal''']]
| align="center" | <math>12T_n+1\,</math>
<math>6n(n+1)+1\,</math>
| align="right" | 1
| align="right" | 13
| align="right" | 37
| align="right" | 73
| align="right" | 121
| align="right" | 181
| align="right" | 253
| align="right" | 337
| align="right" | 433
| align="right" | 541
| align="right" | 661
| align="right" | 793
| align="center" | A003154('''''n'''''+1)
|-
| align="center" | 13
| align="left" | [[Centered tridecagonal numbers|Centered tridecagonal]]
| align="center" | <math>13T_n+1\,</math>
<math>13n(n+1)/2+1\,</math>
| align="right" | 1
| align="right" | 14
| align="right" | 40
| align="right" | 79
| align="right" | 131
| align="right" | 196
| align="right" | 274
| align="right" | 365
| align="right" | 469
| align="right" | 586
| align="right" | 716
| align="right" | 859
| align="center" | A069126('''''n'''''+1)
|-
| align="center" | 14
| align="left" | [[Centered tetradecagonal numbers|Centered tetradecagonal]]
| align="center" | <math>14T_n+1\,</math>
<math>7n(n+1)+1\,</math>
| align="right" | 1
| align="right" | 15
| align="right" | 43
| align="right" | 85
| align="right" | 141
| align="right" | 211
| align="right" | 295
| align="right" | 393
| align="right" | 505
| align="right" | 631
| align="right" | 771
| align="right" | 925
| align="center" | A069127('''''n'''''+1)
|-
| align="center" | '''15'''
| align="left" | [[Centered pentadecagonal numbers|'''Centered pentadecagonal''']]
| align="center" | <math>15T_n+1\,</math>
<math>15n(n+1)/2+1\,</math>
| align="right" | 1
| align="right" | 16
| align="right" | 46
| align="right" | 91
| align="right" | 151
| align="right" | 226
| align="right" | 316
| align="right" | 421
| align="right" | 541
| align="right" | 676
| align="right" | 826
| align="right" | 991
| align="center" | A069128('''''n'''''+1)
|-
| align="center" | '''16'''
| align="left" | [[Centered hexadecagonal numbers|'''Centered hexadecagonal''']]
| align="center" | <math>16T_n+1\,</math>
<math>8n(n+1)+1\,</math>
| align="right" | 1
| align="right" | 17
| align="right" | 49
| align="right" | 97
| align="right" | 161
| align="right" | 241
| align="right" | 337
| align="right" | 449
| align="right" | 577
| align="right" | 721
| align="right" | 881
| align="right" | 1057
| align="center" | A069129('''''n'''''+1)
|-
| align="center" | '''17'''
| align="left" | [[Centered heptadecagonal numbers|'''Centered heptadecagonal''']]
| align="center" | <math>17T_n+1\,</math>
<math>17n(n+1)/2+1\,</math>
| align="right" | 1
| align="right" | 18
| align="right" | 52
| align="right" | 103
| align="right" | 171
| align="right" | 256
| align="right" | 358
| align="right" | 477
| align="right" | 613
| align="right" | 766
| align="right" | 936
| align="right" | 1123
| align="center" | A069130('''''n'''''+1)
|-
| align="center" | 18
| align="left" | [[Centered octadecagonal numbers|Centered octadecagonal]]
| align="center" | <math>18T_n+1\,</math>
<math>9n(n+1)+1\,</math>
| align="right" | 1
| align="right" | 19
| align="right" | 55
| align="right" | 109
| align="right" | 181
| align="right" | 271
| align="right" | 379
| align="right" | 505
| align="right" | 649
| align="right" | 811
| align="right" | 991
| align="right" | 1189
| align="center" | A069131('''''n'''''+1)
|-
| align="center" | 19
| align="left" | [[Centered nonadecagonal numbers|Centered nonadecagonal]]
| align="center" | <math>19T_n+1\,</math>
<math>19n(n+1)/2+1\,</math>
| align="right" | 1
| align="right" | 20
| align="right" | 58
| align="right" | 115
| align="right" | 191
| align="right" | 286
| align="right" | 400
| align="right" | 533
| align="right" | 685
| align="right" | 856
| align="right" | 1046
| align="right" | 1255
| align="center" | A069132('''''n'''''+1)
|-
| align="center" | '''20'''
| align="left" | [[Centered icosagonal numbers|'''Centered icosagonal''']]
| align="center" | <math>20T_n+1\,</math>
<math>10n(n+1)+1\,</math>
| align="right" | 1
| align="right" | 21
| align="right" | 61
| align="right" | 121
| align="right" | 201
| align="right" | 301
| align="right" | 421
| align="right" | 561
| align="right" | 721
| align="right" | 901
| align="right" | 1101
| align="right" | 1321
| align="center" | A069133('''''n'''''+1)
|-
| align="center" | 21
| align="left" | [[Centered icosihenagonal numbers|Centered icosihenagonal]]
| align="center" | <math>21T_n+1\,</math>
<math>21n(n+1)/2+1\,</math>
| align="right" | 1
| align="right" | 22
| align="right" | 64
| align="right" | 127
| align="right" | 211
| align="right" | 316
| align="right" | 442
| align="right" | 589
| align="right" | 757
| align="right" | 946
| align="right" | 1156
| align="right" | 1387
| align="center" | A069178('''''n'''''+1)
|-
| align="center" | 22
| align="left" | [[Centered icosidigonal numbers|Centered icosidigonal]]
| align="center" | <math>22T_n+1\,</math>
<math>11n(n+1)+1\,</math>
| align="right" | 1
| align="right" | 23
| align="right" | 67
| align="right" | 133
| align="right" | 221
| align="right" | 331
| align="right" | 463
| align="right" | 617
| align="right" | 793
| align="right" | 991
| align="right" | 1211
| align="right" | 1453
| align="center" | A069173('''''n'''''+1)
|-
| align="center" | 23
| align="left" | [[Centered icositrigonal numbers|Centered icositrigonal]]
| align="center" | <math>23T_n+1\,</math>
<math>23n(n+1)/2+1\,</math>
| align="right" | 1
| align="right" | 24
| align="right" | 70
| align="right" | 139
| align="right" | 231
| align="right" | 346
| align="right" | 484
| align="right" | 645
| align="right" | 829
| align="right" | 1036
| align="right" | 1266
| align="right" | 1519
| align="center" | A069174('''''n'''''+1)
|-
| align="center" | '''24'''
| align="left" | [[Centered icositetragonal numbers|'''Centered icositetragonal''']]
| align="center" | <math>24T_n+1\,</math>
<math>12n(n+1)+1\,</math>
| align="right" | 1
| align="right" | 25
| align="right" | 73
| align="right" | 145
| align="right" | 241
| align="right" | 361
| align="right" | 505
| align="right" | 673
| align="right" | 865
| align="right" | 1081
| align="right" | 1321
| align="right" | 1585
| align="center" | A069190('''''n'''''+1)
|-
| align="center" | 25
| align="left" | [[Centered icosipentagonal numbers|Centered icosipentagonal]]
| align="center" | <math>25T_n+1\,</math>
<math>25n(n+1)/2+1\,</math>
| align="right" | 1
| align="right" | 26
| align="right" | 76
| align="right" | 151
| align="right" | 251
| align="right" | 376
| align="right" | 526
| align="right" | 701
| align="right" | 901
| align="right" | 1126
| align="right" | 1376
| align="right" | 1651
| align="center" | [[OEIS:A??????]]
|-
| align="center" | 26
| align="left" | [[Centered Icosihexagonal numbers|Centered icosihexagonal]]
| align="center" | <math>26T_n+1\,</math>
<math>13n(n+1)+1\,</math>
| align="right" | 1
| align="right" | 27
| align="right" | 79
| align="right" | 157
| align="right" | 261
| align="right" | 391
| align="right" | 547
| align="right" | 729
| align="right" | 937
| align="right" | 1171
| align="right" | 1431
| align="right" | 1717
| align="center" | [[OEIS:A??????]]
|-
| align="center" | 27
| align="left" | [[Centered icosiheptagonal numbers|Centered icosiheptagonal]]
| align="center" | <math>27T_n+1\,</math>
<math>27n(n+1)/2+1\,</math>
| align="right" | 1
| align="right" | 28
| align="right" | 82
| align="right" | 163
| align="right" | 271
| align="right" | 406
| align="right" | 568
| align="right" | 757
| align="right" | 973
| align="right" | 1216
| align="right" | 1486
| align="right" | 1783
| align="center" | [[OEIS:A??????]]
|-
| align="center" | 28
| align="left" | [[Centered icosioctagonal numbers|Centered icosioctagonal]]
| align="center" | <math>28T_n+1\,</math>
<math>14n(n+1)+1\,</math>
| align="right" | 1
| align="right" | 29
| align="right" | 85
| align="right" | 169
| align="right" | 281
| align="right" | 421
| align="right" | 589
| align="right" | 785
| align="right" | 1009
| align="right" | 1261
| align="right" | 1541
| align="right" | 1849
| align="center" | [[OEIS:A??????]]
|-
| align="center" | 29
| align="left" | [[Centered icosinonagonal numbers|Centered icosinonagonal]]
| align="center" | <math>29T_n+1\,</math>
<math>29n(n+1)/2+1\,</math>
| align="right" | 1
| align="right" | 30
| align="right" | 88
| align="right" | 175
| align="right" | 291
| align="right" | 436
| align="right" | 610
| align="right" | 813
| align="right" | 1045
| align="right" | 1306
| align="right" | 1596
| align="right" | 1915
| align="center" | [[OEIS:A??????]]
|-
| align="center" | '''30'''
| align="left" | [[Centered triacontagonal numbers|'''Centered triacontagonal''']]
| align="center" | <math>30T_n+1\,</math>
<math>15n(n+1)+1\,</math>
| align="right" | 1
| align="right" | 31
| align="right" | 91
| align="right" | 181
| align="right" | 301
| align="right" | 451
| align="right" | 631
| align="right" | 841
| align="right" | 1081
| align="right" | 1351
| align="right" | 1651
| align="right" | 1981
| align="center" | [[OEIS:A??????]]
|-
|}
<br />

== Table of related formulae and values ==

Centered polygonal numbers associated with [[constructible polygons]] (Cf. A003401) (with straightedge and compass) are named in '''bold'''.

<!--== Wikitable color scheme is: [background: #f2f2f2; for table header] [background: #f9f9f9; for table cells] ==-->
{| class="wikitable" align="center" border="1" cellspacing="0" cellpadding="4" style="border-collapse: collapse; border: 1px solid darkgray; background: #f9f9f9; color: black; empty-cells: show; text-align: right;"
|+ '''Centered polygonal numbers related formulae and values'''
|- style="background: #f2f2f2; color: black; text-align: center;"
! width="25" style="text-align: center;" | ''N''<sub>0</sub>
! style="text-align: center;" | Name
! style="text-align: center;" | Generating

function

<math>G_{\{\,_cP^{(2)}_{N_0}(n)\}}(x) =\,</math>


<math>{{x^2+(N_0-2)x+1}\over{(1-x)^3}}\,</math>
! style="text-align: center;" | Order

of basis

<math>g_{\{\,_cP^{(2)}_{N_0}\}}\,</math>
! align="center" | Differences

<math>\,_cP^{(2)}_{N_0}(n) - \,</math>

<math>\,_cP^{(2)}_{N_0}(n-1) =\,</math>


<math>N_0\ n\,</math>
! align="center" | Partial sums

<math>\sum_{n=0}^m {\,_cP^{(2)}_{N_0}(n)} =</math>


<math>N_0 \binom{m+2}{3} + m\,</math>


<math>N_0\ P^{(3)}_{4}(m) + m\,</math>
! align="center" | Partial sums of reciprocals

<math>\sum_{n=0}^m {1\over{\,_cP^{(2)}_{N_0}(n)}} =</math>
! align="center" | Sum of Reciprocals<ref>Downey, Lawrence M., Ong, Boon W., and Sellers, James A., [http://www.math.psu.edu/sellersj/downey_ong_sellers_cmj_preprint.pdf Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers], 2008.</ref>
<math>\sum_{n=0}^\infty{1\over{\,_cP^{(2)}_{N_0}(n)}} =</math>


<math>\scriptstyle {\frac{2\pi}{N_0 \sqrt{1-\tfrac{8}{N_0}}} \tan{\big( \frac{\pi}{2} \sqrt{1-\tfrac{8}{N_0}} \big)}},\,</math>

<math>\scriptstyle N_0 \neq 8,\,</math>

<math>\frac{\pi^2}{8},\ N_0 = 8.\,</math>
|-
| align="center" | '''3'''
| align="left" | [[Centered triangular numbers|'''Centered triangular''']]
| align="center" | <math>{x^2+x+1}\over{(1-x)^3}\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>3n\,</math>
| align="center" | <math>3 \binom{m+2}{3} + m\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>\,</math>
|-
| align="center" | '''4'''
| align="left" | [[Centered square numbers|'''Centered square''']]
| align="center" | <math>{x^2+2x+1}\over{(1-x)^3}\,</math>

<math>{(x+1)^2}\over{(1-x)^3}\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>4n\,</math>
| align="center" | <math>4 \binom{m+2}{3} + m\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>\frac{\pi}{2} \tanh\bigg(\frac{\pi}{2}\bigg)\,</math>
|-
| align="center" | '''5'''
| align="left" | [[Centered pentagonal numbers|'''Centered pentagonal''']]
| align="center" | <math>{x^2+3x+1}\over{(1-x)^3}\,</math> <!-- <ref>[http://www.wolframalpha.com/input/?i=%28x^2%2B%283%29x%2B1%29%2F%281-x%29^3 <math>\scriptstyle {{x^2+3x+1}\over{(1-x)^3}}\,</math>], Wolfram Alpha.</ref> -->
| align="center" | <math>\,</math>
| align="center" | <math>5n\,</math>
| align="center" | <math>5 \binom{m+2}{3} + m\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>\,</math>
|-
| align="center" | '''6'''
| align="left" | [[Centered hexagonal numbers|'''Centered hexagonal''']]
| align="center" | <math>{x^2+4x+1}\over{(1-x)^3}\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>6n\,</math>
| align="center" | <math>6 \binom{m+2}{3} + m\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>\frac{\pi}{\sqrt{3}} \tanh\bigg(\frac{\pi}{2 \sqrt{3}}\bigg)\,</math>
|-
| align="center" | 7
| align="left" | [[Centered heptagonal numbers|Centered heptagonal]]
| align="center" | <math>{x^2+5x+1}\over{(1-x)^3}\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>7n\,</math>
| align="center" | <math>7 \binom{m+2}{3} + m\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>\frac{2\pi}{\sqrt{7}} \tanh\bigg(\frac{\pi}{2 \sqrt{7}}\bigg)\,</math>
|-
| align="center" | '''8'''
| align="left" | [[Centered octagonal numbers|'''Centered octagonal''']]
| align="center" | <math>{x^2+6x+1}\over{(1-x)^3}\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>8n\,</math>
| align="center" | <math>8 \binom{m+2}{3} + m\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>\frac{\pi^2}{8}\,</math>
|-
| align="center" | 9
| align="left" | [[Centered nonagonal numbers|Centered nonagonal]]
| align="center" | <math>{x^2+7x+1}\over{(1-x)^3}\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>9n\,</math>
| align="center" | <math>9 \binom{m+2}{3} + m\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>\frac{2\pi}{3} \tan\bigg(\frac{\pi}{6}\bigg)\,</math>
|-
| align="center" | '''10'''
| align="left" | [[Centered decagonal numbers|'''Centered decagonal''']]
| align="center" | <math>{x^2+8x+1}\over{(1-x)^3}\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>10n\,</math>
| align="center" | <math>10 \binom{m+2}{3} + m\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>\frac{\pi}{\sqrt{5}} \tan\bigg(\frac{\pi}{2 \sqrt{5}}\bigg)\,</math>
|-
| align="center" | 11
| align="left" | [[Centered hendecagonal numbers|Centered hendecagonal]]
| align="center" | <math>{x^2+9x+1}\over{(1-x)^3}\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>11n\,</math>
| align="center" | <math>11 \binom{m+2}{3} + m\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>\,</math>
|-
| align="center" | '''12'''
| align="left" | [[Centered dodecagonal numbers|'''Centered dodecagonal''']]
| align="center" | <math>{x^2+10x+1}\over{(1-x)^3}\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>12n\,</math>
| align="center" | <math>12 \binom{m+2}{3} + m\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>\frac{\pi}{2\sqrt{3}} \tan\bigg(\frac{\pi}{2 \sqrt{3}}\bigg)\,</math>
|-
| align="center" | 13
| align="left" | [[Centered tridecagonal numbers|Centered tridecagonal]]
| align="center" | <math>{x^2+11x+1}\over{(1-x)^3}\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>13n\,</math>
| align="center" | <math>13 \binom{m+2}{3} + m\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>\,</math>
|-
| align="center" | 14
| align="left" | [[Centered tetradecagonal numbers|Centered tetradecagonal]]
| align="center" | <math>{x^2+12x+1}\over{(1-x)^3}\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>14n\,</math>
| align="center" | <math>14 \binom{m+2}{3} + m\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>\,</math>
|-
| align="center" | '''15'''
| align="left" | [[Centered pentadecagonal numbers|'''Centered pentadecagonal''']]
| align="center" | <math>{x^2+13x+1}\over{(1-x)^3}\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>15n\,</math>
| align="center" | <math>15 \binom{m+2}{3} + m\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>\,</math>
|-
| align="center" | '''16'''
| align="left" | [[Centered hexadecagonal numbers|'''Centered hexadecagonal''']]
| align="center" | <math>{x^2+14x+1}\over{(1-x)^3}\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>16n\,</math>
| align="center" | <math>16 \binom{m+2}{3} + m\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>\frac{\pi}{4\sqrt{2}} \tan\bigg(\frac{\pi}{2 \sqrt{2}}\bigg)\,</math>
|-
| align="center" | '''17'''
| align="left" | [[Centered heptadecagonal numbers|'''Centered heptadecagonal''']]
| align="center" | <math>{x^2+15x+1}\over{(1-x)^3}\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>17n\,</math>
| align="center" | <math>17 \binom{m+2}{3} + m\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>\,</math>
|-
| align="center" | 18
| align="left" | [[Centered octadecagonal numbers|Centered octadecagonal]]
| align="center" | <math>{x^2+16x+1}\over{(1-x)^3}\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>18n\,</math>
| align="center" | <math>18 \binom{m+2}{3} + m\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>\,</math>
|-
| align="center" | 19
| align="left" | [[Centered nonadecagonal numbers|Centered nonadecagonal]]
| align="center" | <math>{x^2+17x+1}\over{(1-x)^3}\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>19n\,</math>
| align="center" | <math>19 \binom{m+2}{3} + m\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>\,</math>
|-
| align="center" | '''20'''
| align="left" | [[Centered icosagonal numbers|'''Centered icosagonal''']]
| align="center" | <math>{x^2+18x+1}\over{(1-x)^3}\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>20n\,</math>
| align="center" | <math>20 \binom{m+2}{3} + m\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>\,</math>
|-
| align="center" | 21
| align="left" | [[Centered icosihenagonal numbers|Centered icosihenagonal]]
| align="center" | <math>{x^2+19x+1}\over{(1-x)^3}\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>21n\,</math>
| align="center" | <math>21 \binom{m+2}{3} + m\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>\,</math>
|-
| align="center" | 22
| align="left" | [[Centered icosidigonal numbers|Centered icosidigonal]]
| align="center" | <math>{x^2+20x+1}\over{(1-x)^3}\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>22n\,</math>
| align="center" | <math>22 \binom{m+2}{3} + m\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>\,</math>
|-
| align="center" | 23
| align="left" | [[Centered icositrigonal numbers|Centered icositrigonal]]
| align="center" | <math>{x^2+21x+1}\over{(1-x)^3}\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>23n\,</math>
| align="center" | <math>23 \binom{m+2}{3} + m\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>\,</math>
|-
| align="center" | '''24'''
| align="left" | [[Centered icositetragonal numbers|'''Centered icositetragonal''']]
| align="center" | <math>{x^2+22x+1}\over{(1-x)^3}\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>24n\,</math>
| align="center" | <math>24 \binom{m+2}{3} + m\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>\frac{\pi}{4\sqrt{6}} \tan\bigg( \frac{\pi}{\sqrt{6}} \bigg)\,</math>
|-
| align="center" | 25
| align="left" | [[Centered icosipentagonal numbers|Centered icosipentagonal]]
| align="center" | <math>{x^2+23x+1}\over{(1-x)^3}\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>25n\,</math>
| align="center" | <math>25 \binom{m+2}{3} + m\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>\,</math>
|-
| align="center" | 26
| align="left" | [[Centered icosihexagonal numbers|Centered icosihexagonal]]
| align="center" | <math>{x^2+24x+1}\over{(1-x)^3}\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>26n\,</math>
| align="center" | <math>26 \binom{m+2}{3} + m\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>\,</math>
|-
| align="center" | 27
| align="left" | [[Centered icosiheptagonal numbers|Centered icosiheptagonal]]
| align="center" | <math>{x^2+25x+1}\over{(1-x)^3}\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>27n\,</math>
| align="center" | <math>27 \binom{m+2}{3} + m\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>\,</math>
|-
| align="center" | 28
| align="left" | [[Centered icosioctagonal numbers|Centered icosioctagonal]]
| align="center" | <math>{x^2+26x+1}\over{(1-x)^3}\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>28n\,</math>
| align="center" | <math>28 \binom{m+2}{3} + m\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>\,</math>
|-
| align="center" | 29
| align="left" | [[Centered icosinonagonal numbers|Centered icosinonagonal]]
| align="center" | <math>{x^2+27x+1}\over{(1-x)^3}\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>29n\,</math>
| align="center" | <math>29 \binom{m+2}{3} + m\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>\,</math>
|-
| align="center" | '''30'''
| align="left" | [[Centered triacontagonal numbers|'''Centered triacontagonal''']]
| align="center" | <math>{x^2+28x+1}\over{(1-x)^3}\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>30n\,</math>
| align="center" | <math>30 \binom{m+2}{3} + m\,</math>
| align="center" | <math>\,</math>
| align="center" | <math>\frac{\pi}{6\sqrt{5}} \tan\bigg(\frac{\pi}{\sqrt{5}}\bigg)\,</math>
|-
|}
<br />

== Table of sequences ==

<!--== Wikitable color scheme is: [background: #f2f2f2; for table header] [background: #f9f9f9; for table cells] ==-->
{| class="wikitable" align="center" border="1" cellspacing="0" cellpadding="4" style="border-collapse: collapse; border: 1px solid darkgray; background: #f9f9f9; color: black; empty-cells: show; text-align: left;"
|+ '''Centered polygonal numbers sequences'''
|- style="background: #f2f2f2; color: black; text-align: center;"
! width="25" style="text-align: center;" | ''N''<sub>0</sub>
! style="text-align: center;" | <math>\,_cP^{(2)}_{N_0}(n),\ n \ge 0</math> sequences
|-
| style="text-align: center;" | '''3'''
| {1, 4, 10, 19, 31, 46, 64, 85, 109, 136, 166, 199, 235, 274, 316, 361, 409, 460, 514, 571, 631, 694, 760, 829, 901, 976, 1054, 1135, 1219, 1306, 1396, 1489, 1585, 1684, ...}
|-
| style="text-align: center;" | '''4'''
| {1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 265, 313, 365, 421, 481, 545, 613, 685, 761, 841, 925, 1013, 1105, 1201, 1301, 1405, 1513, 1625, 1741, 1861, 1985, 2113, ...}
|-
| style="text-align: center;" | '''5'''
| {1, 6, 16, 31, 51, 76, 106, 141, 181, 226, 276, 331, 391, 456, 526, 601, 681, 766, 856, 951, 1051, 1156, 1266, 1381, 1501, 1626, 1756, 1891, 2031, 2176, 2326, 2481, 2641, ...}
|-
| style="text-align: center;" | '''6'''
| {1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, 469, 547, 631, 721, 817, 919, 1027, 1141, 1261, 1387, 1519, 1657, 1801, 1951, 2107, 2269, 2437, 2611, 2791, 2977, 3169, ...}
|-
| style="text-align: center;" | '''7'''
| {1, 8, 22, 43, 71, 106, 148, 197, 253, 316, 386, 463, 547, 638, 736, 841, 953, 1072, 1198, 1331, 1471, 1618, 1772, 1933, 2101, 2276, 2458, 2647, 2843, 3046, 3256, 3473, ...}
|-
| style="text-align: center;" | '''8'''
| {1, 9, 25, 49, 81, 121, 169, 225, 289, 361, 441, 529, 625, 729, 841, 961, 1089, 1225, 1369, 1521, 1681, 1849, 2025, 2209, 2401, 2601, 2809, 3025, 3249, 3481, 3721, 3969, ...}
|-
| style="text-align: center;" | '''9'''
| {1, 10, 28, 55, 91, 136, 190, 253, 325, 406, 496, 595, 703, 820, 946, 1081, 1225, 1378, 1540, 1711, 1891, 2080, 2278, 2485, 2701, 2926, 3160, 3403, 3655, 3916, 4186, 4465, ...}
|-
| style="text-align: center;" | '''10'''
| {1, 11, 31, 61, 101, 151, 211, 281, 361, 451, 551, 661, 781, 911, 1051, 1201, 1361, 1531, 1711, 1901, 2101, 2311, 2531, 2761, 3001, 3251, 3511, 3781, 4061, 4351, 4651, ...}
|-
| style="text-align: center;" | '''11'''
| {1, 12, 34, 67, 111, 166, 232, 309, 397, 496, 606, 727, 859, 1002, 1156, 1321, 1497, 1684, 1882, 2091, 2311, 2542, 2784, 3037, 3301, 3576, 3862, 4159, 4467, 4786, ...}
|-
| style="text-align: center;" | '''12'''
| {1, 13, 37, 73, 121, 181, 253, 337, 433, 541, 661, 793, 937, 1093, 1261, 1441, 1633, 1837, 2053, 2281, 2521, 2773, 3037, 3313, 3601, 3901, 4213, 4537, 4873, 5221, 5581, ...}
|-
| style="text-align: center;" | '''13'''
| {1, 14, 40, 79, 131, 196, 274, 365, 469, 586, 716, 859, 1015, 1184, 1366, 1561, 1769, 1990, 2224, 2471, 2731, 3004, 3290, 3589, 3901, 4226, 4564, 4915, 5279, 5656, 6046, ...}
|-
| style="text-align: center;" | '''14'''
| {1, 15, 43, 85, 141, 211, 295, 393, 505, 631, 771, 925, 1093, 1275, 1471, 1681, 1905, 2143, 2395, 2661, 2941, 3235, 3543, 3865, 4201, 4551, 4915, 5293, 5685, 6091, 6511, ...}
|-
| style="text-align: center;" | '''15'''
| {1, 16, 46, 91, 151, 226, 316, 421, 541, 676, 826, 991, 1171, 1366, 1576, 1801, 2041, 2296, 2566, 2851, 3151, 3466, 3796, 4141, 4501, 4876, 5266, 5671, 6091, 6526, 6976, ...}
|-
| style="text-align: center;" | '''16'''
| {1, 17, 49, 97, 161, 241, 337, 449, 577, 721, 881, 1057, 1249, 1457, 1681, 1921, 2177, 2449, 2737, 3041, 3361, 3697, 4049, 4417, 4801, 5201, 5617, 6049, 6497, 6961, 7441, ...}
|-
| style="text-align: center;" | '''17'''
| {1, 18, 52, 103, 171, 256, 358, 477, 613, 766, 936, 1123, 1327, 1548, 1786, 2041, 2313, 2602, 2908, 3231, 3571, 3928, 4302, 4693, 5101, 5526, 5968, 6427, 6903, 7396, 7906, ...}
|-
| style="text-align: center;" | '''18'''
| {1, 19, 55, 109, 181, 271, 379, 505, 649, 811, 991, 1189, 1405, 1639, 1891, 2161, 2449, 2755, 3079, 3421, 3781, 4159, 4555, 4969, 5401, 5851, 6319, 6805, 7309, 7831, 8371, ...}
|-
| style="text-align: center;" | '''19'''
| {1, 20, 58, 115, 191, 286, 400, 533, 685, 856, 1046, 1255, 1483, 1730, 1996, 2281, 2585, 2908, 3250, 3611, 3991, 4390, 4808, 5245, 5701, 6176, 6670, 7183, 7715, 8266, 8836, ...}
|-
| style="text-align: center;" | '''20'''
| {1, 21, 61, 121, 201, 301, 421, 561, 721, 901, 1101, 1321, 1561, 1821, 2101, 2401, 2721, 3061, 3421, 3801, 4201, 4621, 5061, 5521, 6001, 6501, 7021, 7561, 8121, 8701, 9301, ...}
|-
| style="text-align: center;" | '''21'''
| {1, 22, 64, 127, 211, 316, 442, 589, 757, 946, 1156, 1387, 1639, 1912, 2206, 2521, 2857, 3214, 3592, 3991, 4411, 4852, 5314, 5797, 6301, 6826, 7372, 7939, 8527, 9136, 9766, ...}
|-
| style="text-align: center;" | '''22'''
| {1, 23, 67, 133, 221, 331, 463, 617, 793, 991, 1211, 1453, 1717, 2003, 2311, 2641, 2993, 3367, 3763, 4181, 4621, 5083, 5567, 6073, 6601, 7151, 7723, 8317, 8933, 9571, 10231, ...}
|-
| style="text-align: center;" |'''23'''
| {1, 24, 70, 139, 231, 346, 484, 645, 829, 1036, 1266, 1519, 1795, 2094, 2416, 2761, 3129, 3520, 3934, 4371, 4831, 5314, 5820, 6349, 6901, 7476, 8074, 8695, 9339, 10006, ...}
|-
| style="text-align: center;" | '''24'''
| {1, 25, 73, 145, 241, 361, 505, 673, 865, 1081, 1321, 1585, 1873, 2185, 2521, 2881, 3265, 3673, 4105, 4561, 5041, 5545, 6073, 6625, 7201, 7801, 8425, 9073, 9745, 10441, ...}
|-
| style="text-align: center;" | '''25'''
| {1, 26, 76, 151, 251, 376, 526, 701, 901, 1126, 1376, 1651, 1951, 2276, 2626, 3001, 3401, 3826, 4276, 4751, 5251, 5776, 6326, 6901, 7501, 8126, 8776, 9451, 10151, 10876, ...}
|-
| style="text-align: center;" | '''26'''
| {1, 27, 79, 157, 261, 391, 547, 729, 937, 1171, 1431, 1717, 2029, 2367, 2731, 3121, 3537, 3979, 4447, 4941, 5461, 6007, 6579, 7177, 7801, 8451, 9127, 9829, 10557, 11311, ...}
|-
| style="text-align: center;" | '''27'''
| {1, 28, 82, 163, 271, 406, 568, 757, 973, 1216, 1486, 1783, 2107, 2458, 2836, 3241, 3673, 4132, 4618, 5131, 5671, 6238, 6832, 7453, 8101, 8776, 9478, 10207, 10963, 11746, ...}
|-
| style="text-align: center;" | '''28'''
| {1, 29, 85, 169, 281, 421, 589, 785, 1009, 1261, 1541, 1849, 2185, 2549, 2941, 3361, 3809, 4285, 4789, 5321, 5881, 6469, 7085, 7729, 8401, 9101, 9829, 10585, 11369, 12181, ...}
|-
| style="text-align: center;" | '''29'''
| {1, 30, 88, 175, 291, 436, 610, 813, 1045, 1306, 1596, 1915, 2263, 2640, 3046, 3481, 3945, 4438, 4960, 5511, 6091, 6700, 7338, 8005, 8701, 9426, 10180, 10963, 11775, 12616, ...}
|-
| style="text-align: center;" | '''30'''
| {1, 31, 91, 181, 301, 451, 631, 841, 1081, 1351, 1651, 1981, 2341, 2731, 3151, 3601, 4081, 4591, 5131, 5701, 6301, 6931, 7591, 8281, 9001, 9751, 10531, 11341, 12181, 13051, ...}
|-
|}
<br>


==References==
==References==

Revision as of 05:39, 2 July 2019

The centered polygonal numbers are a class of series of figurate numbers, each formed by a central dot, surrounded by polygonal layers with a constant number of sides. Each side of a polygonal layer contains one dot more than a side in the previous layer, so starting from the second polygonal layer each layer of a centered k-gonal number contains k more points than the previous layer.

Examples

Each sequence is a multiple of the triangular numbers plus 1. For example, the centered square numbers are four times the triangular numbers plus 1.

These series consist of the

and so on.

The following diagrams show a few examples of centered polygonal numbers and their geometric construction. Compare these diagrams with the diagrams in Polygonal number.

centered
triangular
number
centered
square
number
centered
pentagonal
number
centered
hexagonal
number

Centered square numbers

1     5     13     25
   

   



   





Centered hexagonal numbers

1             7             19                  37
* **
***
**
***
****
*****
****
***
****
*****
******
*******
******
*****
****

Formula

As can be seen in the above diagrams, the nth centered k-gonal number can be obtained by placing k copies of the (n−1)th triangular number around a central point; therefore, the nth centered k-gonal number can be mathematically represented by

The difference of the n-th and the (n+1)-th consecutive centered k-gonal numbers is k(2n+1).

Just as is the case with regular polygonal numbers, the first centered k-gonal number is 1. Thus, for any k, 1 is both k-gonal and centered k-gonal. The next number to be both k-gonal and centered k-gonal can be found using the formula:

which tells us that 10 is both triangular and centered triangular, 25 is both square and centered square, etc.

Whereas a prime number p cannot be a polygonal number (except the trivial case, i.e. each p is the second p-gonal number), many centered polygonal numbers are primes. In fact, if k ≥ 3, k ≠ 8, k ≠ 9, then there are infinitely many centered k-gonal numbers which are primes (assuming the Bunyakovsky conjecture). (Since all centered octagonal numbers are also square numbers, and all centered nonagonal numbers are also triangular numbers (and not equal to 3), thus both of them cannot be prime numbers)

Sum of Reciprocals

The sum of reciprocals for the centered k-gonal numbers is[1]

, if k ≠ 8
, if k = 8

References

  • Neil Sloane & Simon Plouffe (1995). The Encyclopedia of Integer Sequences. San Diego: Academic Press.: Fig. M3826
  • Weisstein, Eric W. "Centered polygonal number". MathWorld.
  • F. Tapson (1999). The Oxford Mathematics Study Dictionary (2nd ed.). Oxford University Press. pp. 88–89. ISBN 0-19-914-567-9.