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In mathematics, a polygonal number is a number represented as dots or pebbles arrayed in the shape of a polygon. The dots were thought of as alphas (units). These are one type of figurate numbers.

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

GrayDotX
GrayDotXGrayDotX
GrayDotXGrayDotXGrayDotX
GrayDotXGrayDotXGrayDotXGrayDotX

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

GrayDotGrayDotGrayDot
GrayDotGrayDotGrayDot
GrayDotGrayDotGrayDot

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

GrayDotGrayDotGrayDotGrayDotGrayDotGrayDot
GrayDotGrayDotGrayDotGrayDotGrayDotGrayDot
GrayDotGrayDotGrayDotGrayDotGrayDotGrayDot
GrayDotGrayDotGrayDotGrayDotGrayDotGrayDot
GrayDotGrayDotGrayDotGrayDotGrayDotGrayDot
GrayDotGrayDotGrayDotGrayDotGrayDotGrayDot
GrayDotX
GrayDotXGrayDotX
GrayDotXGrayDotXGrayDotX
GrayDotXGrayDotXGrayDotXGrayDotX
GrayDotXGrayDotXGrayDotXGrayDotXGrayDotX
GrayDotXGrayDotXGrayDotXGrayDotXGrayDotXGrayDotX
GrayDotXGrayDotXGrayDotXGrayDotXGrayDotXGrayDotXGrayDotX
GrayDotXGrayDotXGrayDotXGrayDotXGrayDotXGrayDotXGrayDotXGrayDotX

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
1 3 6 10
RedDotX GrayDotX
RedDotXRedDotX
GrayDotX
GrayDotXGrayDotX
RedDotXRedDotXRedDotX
GrayDotX
GrayDotXGrayDotX
GrayDotXGrayDotXGrayDotX
RedDotXRedDotXRedDotXRedDotX
Square numbers
1 4 9 16
RedDot GrayDotRedDot
RedDotRedDot
GrayDotGrayDotRedDot
GrayDotGrayDotRedDot
RedDotRedDotRedDot
GrayDotGrayDotGrayDotRedDot
GrayDotGrayDotGrayDotRedDot
GrayDotGrayDotGrayDotRedDot
RedDotRedDotRedDotRedDot

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 regular lattice like above. For example, the first few hexagonal numbers are:

1 6 15 28
RedDotX RedDotRedDotX
GrayDotXBlank300RedDotX
RedDotXRedDotX
RedDotRedDotXRedDotX
GrayDotXGrayDotXBlank300RedDotX
GrayDotXBlank300GrayDotXBlank300RedDotX
GrayDotXGrayDotXBlank300RedDotX
RedDotRedDotXRedDotX
RedDotRedDotRedDotXRedDotX
GrayDotXGrayDotXGrayDotXBlank300RedDotX
GrayDotXGrayDotXBlank300GrayDotXBlank300RedDotX
GrayDotXBlank300GrayDotXBlank300GrayDotXBlank300RedDotX
GrayDotXGrayDotXBlank300GrayDotXBlank300RedDotX
GrayDotXGrayDotXGrayDotXBlank300RedDotX
RedDotRedDotRedDotXRedDotX

If s is the number of sides in a polygon, the formula for the nth s-gonal number is \frac{n^2[s-2]-n[s-4]}{2}.

NameFormulan=12345678910111213
Triangular½(n² + n) 13610152128364555667891
Squaren² 149162536496481100121144169
Pentagonal½(3n² - n) 15122235517092117145176210247
Hexagonal2n² - n 161528456691120153190231276325
Heptagonal½(5n² - 3n) 1718345581112148189235286342403
Octagonal3n² - 2n 1821406596133176225280341408481
Nonagonal½(7n² - 5n) 19244675111154204261325396474559
Decagonal(2n)² - 3n 110275285126175232297370451540637
Hendecagonal½((3n)² - 7n) 111305895141196260333415506606715
Dodecagonal5n² - 4n 1123364105156217288369460561672793
Tridecagonal½(11n² - 9n) 1133670115171238316405505616738871
Tetradecagonal6n² - 5n 1143976125186259344441550671804949
Pentadecagonal½(13n² - 11n) 11542821352012803724775957268701027
Hexadecagonal7n² - 6n 11645881452163014005136407819361105
Heptadecagonal½(15n² - 13n) 117489415523132242854968583610021183
Octadecagonal8n² - 7n 1185110016524634345658573089110681261
Nonadecagonal½(17n² - 15n) 1195410617526136448462177594611341339
Icosagonal(3n)² - 8n 12057112185276385512657820100112001417
Icosihenagonal½(19n² - 17n) 12160118195291406540693865105612661495
Icosidigonal10n² - 9n 12263124205306427568729910111113321573
Icositrigonal½(21n² - 19n) 12366130215321448596765955116613981651
Icositetragonal11n² - 10n 124691362253364696248011000122114641729
Icosipentagonal½(23n² - 21n) 125721422353514906528371045127615301807
Icosihexagonal12n² - 11n 126751482453665116808731090133115961885
Icosiheptagonal½((5n)² - 23n) 127781542553815327089091135138616621963
Icosioctagonal13n² - 12n 128811602653965537369451180144117282041
Icosinonagonal½(27n² - 25n) 129841662754115747649811225149617942119
Triacontagonal14n² - 13n 1308717228542659579210171270155118602197

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").

For a given s-gonal number x, one can find n by

n = \frac{\sqrt{8sx+s^2-16x-8s+16}+s-4}{2s-4}.

References

External links


ar:عدد مضلعيio:Poligona nombro it:Numero poligonale he:מספר מצולע hu:Sokszögszámok nl:Veelhoeksgetalsl:Mnogokotniško številota:பல்கோண எண்

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