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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.
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| == Definition and examples ==
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| The number 10, for example, can be arranged as a [[triangle]] (see [[triangular number]]):
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| :{|
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| | 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|*]]
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| |}
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| But 10 cannot be arranged as a [[square (geometry)|square]]. The number 9, on the other hand, can be (see [[square number]]):
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| :{|
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| | 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|*]]
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| |}
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| Some numbers, like 36, can be arranged both as a square and as a triangle (see [[square triangular number]]):
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| :{|
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| |- align="center" valign="bottom"
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| |[[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|*]]
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| |[[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|*]]
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| |}
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| 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.
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| ===Triangular numbers===
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| [[File:Polygonal Number 3.gif|500px|none]]
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| <br>
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| ===Square numbers===
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| [[File:Polygonal Number 4.gif|500px|none]]
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| <br>
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| 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.
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| ===Pentagonal numbers===
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| [[File:Polygonal Number 5.gif|500px|none]]
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| <br>
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| ===Hexagonal numbers===
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| [[File:Polygonal Number 6.gif|500px|none]]
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| <br>
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| ==Formula==
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| 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
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| :<math>P(s,n) = \frac{n^2(s-2)-n(s-4)}{2}</math>
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| or
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| :<math>P(s,n) = \frac{n(s-2)(n-1)}{2}+n</math>
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| The ''n''<sup>th</sup> ''s''-gonal number is also related to the triangular numbers ''T''<sub>''n''</sub> as follows:
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| :<math>P(s,n) = (s-2)T_{n-1} + n = (s-3)T_{n-1} + T_n\, .</math>
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| Thus:
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| :<math>P(s,n+1)-P(s,n) = (s-2)n + 1\, ,</math>
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| :<math>P(s+1,n) - P(s,n) = T_{n-1} = \frac{n(n-1)}{2}\, .</math>
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| For a given ''s''-gonal number ''P''(''s'',''n'') = ''x'', one can find ''n'' by
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| :<math>n = \frac{\sqrt{(8s-16)x+(s-4)^2}+s-4}{2s-4}.</math>
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| ==Table of values==
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| {| class="wikitable" border="1"
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| |-
| |
| ! s
| |
| ! Name
| |
| ! Formula
| |
| ! align="right" | ''n'' = 1
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| ! align="right" | ''n'' = 2
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| ! align="right" | ''n'' = 3
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| ! align="right" | ''n'' = 4
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| ! align="right" | ''n'' = 5
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| ! align="right" | ''n'' = 6
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| ! align="right" | ''n'' = 7
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| ! align="right" | ''n'' = 8
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| ! align="right" | ''n'' = 9
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| ! align="right" | ''n'' = 10
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| ! 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>
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| ! align="center" | [[On-Line Encyclopedia of Integer Sequences|OEIS]] number
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| |-
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| | align="right" | 3
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| | [[Triangular number|Triangular]]
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| | ½(''n''²+''n'')
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| | align="right" | 1
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| | align="right" | 3
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| | align="right" | 6
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| | align="right" | 10
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| | align="right" | 15
| |
| | align="right" | 21
| |
| | align="right" | 28
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| | align="right" | 36
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| | align="right" | 45
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| | align="right" | 55
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| ! align="right" | <math>{2}</math>
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| | {{OEIS link|id=A000217}}
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| |-
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| | align="right" | 4
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| | [[Square number|Square]]
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| | ''n''²
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| | align="right" | 1
| |
| | align="right" | 4
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| | align="right" | 9
| |
| | align="right" | 16
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| | 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>
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| | {{OEIS link|id=A000290}}
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| |-
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| | align="right" | 5
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| | [[Pentagonal number|Pentagonal]]
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| | ½(3''n''² - ''n'')
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| | align="right" | 1
| |
| | align="right" | 5
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| | align="right" | 12
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| | 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>
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| | {{OEIS link|id=A000326}}
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| |-
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| | align="right" | 6
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| | [[Hexagonal number|Hexagonal]]
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| | ½(4''n''² - 2''n'')
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| | align="right" | 1
| |
| | align="right" | 6
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| | align="right" | 15
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| | align="right" | 28
| |
| | align="right" | 45
| |
| | align="right" | 66
| |
| | align="right" | 91
| |
| | align="right" | 120
| |
| | align="right" | 153
| |
| | align="right" | 190
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| ! align="right" | <math>{ 2\ln\left(2\right) }</math>
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| | {{OEIS link|id=A000384}}
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| |-
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| | align="right" | 7
| |
| | [[Heptagonal number|Heptagonal]]
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| | ½(5''n''² - 3''n'')
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| | align="right" | 1
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| | align="right" | 7
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| | align="right" | 18
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| | align="right" | 34
| |
| | align="right" | 55
| |
| | align="right" | 81
| |
| | align="right" | 112
| |
| | align="right" | 148
| |
| | align="right" | 189
| |
| | align="right" | 235
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| ! align="right" | <math>\begin{matrix}
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| \frac{1}{15}{\pi}{\sqrt{25-10\sqrt{5}}}+\frac{2}{3}\ln(5) \\
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| +\frac{{1}+\sqrt{5}}{3}\ln\left(\frac{1}{2}\sqrt{10-2\sqrt{5}}\right) \\
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| +\frac{{1}-\sqrt{5}}{3}\ln\left(\frac{1}{2}\sqrt{10+2\sqrt{5}}\right)
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| \end{matrix}</math><ref>http://www.siam.org/journals/problems/downloadfiles/07-003s.pdf</ref>
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| | {{OEIS link|id=A000566}}
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| |-
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| | align="right" | 8
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| | [[Octagonal number|Octagonal]]
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| | ½(6''n''² - 4''n'')
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| | align="right" | 1
| |
| | align="right" | 8
| |
| | align="right" | 21
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| | 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>
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| | {{OEIS link|id=A000567}}
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| |-
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| | align="right" | 9
| |
| | [[Nonagonal number|Nonagonal]]
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| | ½(7''n''² - 5''n'')
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| | align="right" | 1
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| | align="right" | 9
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| | align="right" | 24
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| | align="right" | 46
| |
| | align="right" | 75
| |
| | align="right" | 111
| |
| | align="right" | 154
| |
| | align="right" | 204
| |
| | align="right" | 261
| |
| | align="right" | 325
| |
| ! align="right" |
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| | {{OEIS link|id=A001106}}
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| |-
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| | align="right" | 10
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| | [[Decagonal number|Decagonal]]
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| | ½(8''n''² - 6''n'')
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| | align="right" | 1
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| | align="right" | 10
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| | align="right" | 27
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| | align="right" | 52
| |
| | align="right" | 85
| |
| | align="right" | 126
| |
| | align="right" | 175
| |
| | align="right" | 232
| |
| | align="right" | 297
| |
| | align="right" | 370
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| ! align="right" | <math>{ {\ln\left(2\right)} + {\pi\over6} }</math>
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| | {{OEIS link|id=A001107}}
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| |-
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| | 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}}
| |
| |-
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| | 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
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| ! align="right" | <math>{ {2\ln\left(2\right)\over5} + {3\ln\left(3\right)\over 10} + {\pi\sqrt{3}\over10} }</math>
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| | {{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]]
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| | |
| ==Notes==
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| {{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}}
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| | |
| ==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]
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| *{{youtube|id=YOiZ459lZ7A|title=Polygonal Numbers on the Ulam Spiral grid}}
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| | |
| {{Classes of natural numbers}}
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| [[Category:Figurate numbers]]
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| [[Category:Recreational mathematics]]
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| | |
| [[ru:Последовательность двенадцатиугольника]]
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