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| Line 1: |
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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.
| | Msvcr71.dll is an important file that assists help Windows procedure different components of the program including significant files. Specifically, the file is employed to aid run corresponding files in the "Virtual C Runtime Library". These files are important in accessing any settings which help the different applications plus programs inside the system. The msvcr71.dll file fulfills countless significant functions; nonetheless it's not spared from getting damaged or corrupted. Once the file gets corrupted or damaged, the computer will have a hard time processing and reading components of the program. Nonetheless, consumers need not panic considering this problem will be solved by following several procedures. And I can show we several tricks regarding Msvcr71.dll.<br><br>Your PC registry starts to receive mistakes and fragmented the more we employ the computer considering we enter more information each time, because well as make changes in the systems plus setup. When the registry starts to get overloaded and full of errors, a computer can eventually crash. It can be done to fix it on your yet quite dangerous, specifically in the event you have no extensive experience in doing this. Therefore, do NOT even attempt to do this oneself.<br><br>System tray icon makes it convenient to launch the system plus displays "clean" status or the amount of errors in the last scan. The ability to locate and remove the Invalid class keys plus shell extensions is one of the leading blessings of the system. That is not routine function for the alternative Registry Cleaners. Class keys plus shell extensions which are not functioning can seriously slow down the computer. RegCure scans to obtain invalid entries and delete them.<br><br>Always see with it which we have installed antivirus, anti-spyware plus anti-adware programs and have them up-to-date on a regular basis. This can help stop windows XP running slow.<br><br>So to fix this, we simply should be capable to create all registry files non-corrupted again. This might dramatically accelerate the loading time of the computer plus will allow we to do a big amount of items on it again. And fixing these files couldn't be simpler - we just have to utilize a tool called a [http://bestregistrycleanerfix.com/registry-mechanic pc tools registry mechanic].<br><br>Another key element whenever we compare registry products is having a facility to manage your start-up tasks. This just means to select what programs you want to commence whenever we commence your PC. If you have unwanted programs beginning whenever you boot up a PC this might cause a slow running computer.<br><br>Your disk needs space in order to run smoothly. By freeing up several area from a disk, you'll be capable to speed up a PC a bit. Delete all file inside the temporary internet files folder, recycle bin, well-defined shortcuts plus icons from your desktop which you never utilize plus remove programs we never use.<br><br>There is a lot a superior registry cleaner can do for the computer. It could check for plus download updates for Windows, Java and Adobe. Keeping updates present is an significant piece of superior computer wellness. It may also protect your personal and company confidentiality also as your online protection. |
| | |
| == Definition and examples ==
| |
| | |
| The number 10, for example, can be arranged as a [[triangle]] (see [[triangular number]]):
| |
| | |
| :{|
| |
| | 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|*]]
| |
| |}
| |
| | |
| But 10 cannot be arranged as a [[square (geometry)|square]]. The number 9, on the other hand, can be (see [[square number]]):
| |
| | |
| :{|
| |
| | 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|*]]
| |
| |}
| |
| | |
| Some numbers, like 36, can be arranged both as a square and as a triangle (see [[square triangular number]]):
| |
| | |
| :{|
| |
| |- 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.
| |
| | |
| ===Triangular numbers===
| |
| [[File:Polygonal Number 3.gif|500px|none]]
| |
| <br> | |
| | |
| ===Square numbers===
| |
| | |
| [[File:Polygonal Number 4.gif|500px|none]]
| |
| <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.
| |
| | |
| ===Pentagonal numbers===
| |
| | |
| [[File:Polygonal Number 5.gif|500px|none]]
| |
| <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) = \frac{n(s-2)(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{(8s-16)x+(s-4)^2}+s-4}{2s-4}.</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
| |
| | 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{1}{15}{\pi}{\sqrt{25-10\sqrt{5}}}+\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)
| |
| \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" |
| |
| | {{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").
| |
| | |
| ==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, …
| |
| | {{OEIS link|id=A001110}}
| |
| |-
| |
| | 5
| |
| | 3
| |
| | 1, 210, 40755, 7906276, …
| |
| | {{OEIS link|id=A014979}}
| |
| |-
| |
| | 5
| |
| | 4
| |
| | 1, 9801, 94109401, …
| |
| | {{OEIS link|id=A036353}}
| |
| |-
| |
| | 6
| |
| | 3
| |
| | All hexagonal numbers are also triangular.
| |
| | {{OEIS link|id=A000384}}
| |
| |-
| |
| | 6
| |
| | 4
| |
| | 1, 1225, 1413721, 1631432881, …
| |
| | {{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, …
| |
| | {{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, …
| |
| | {{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, …
| |
| | {{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 ''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 such number has yet to appear in print.<ref>{{MathWorld|title=Pentagonal Square Triangular Number | urlname=PentagonalSquareTriangularNumber}}</ref> All hexagonal square numbers are also hexagonal square triangular numbers, and 1225 is actually a hecticositetragonal, hexacontagonal, icosinonagonal, hexagonal, square, triangular number.
| |
| | |
| ==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}}
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| | |
| {{Classes of natural numbers}}
| |
| [[Category:Figurate numbers]]
| |
| [[Category:Recreational mathematics]]
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| | |
| [[ru:Последовательность двенадцатиугольника]]
| |