Polygonal number

In mathematics, a polygonal number is a number that can be arranged as a regular polygon. Ancient mathematicians discovered that numbers could be arranged in certain ways when they were represented by pebbles or seeds; such numbers, which can be made from figures, are generally called figurate numbers.

Related Topics:
Mathematics - Number - Polygon - Mathematician - Figurate number

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

Related Topics:
Triangle - Triangular number

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x

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x x

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x x x

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x x x x

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

Related Topics:
Square - Square number

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x x x

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x x x

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x x x

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

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x x x x x x

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x x x x x x

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x x x x x x

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x x x x x x

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x x x x x x

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x x x x x x

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x

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x x

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x x x

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x x x x

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x x x x x

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x x x x x x

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x x x x x x x

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x x x x x x x x

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The method 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 +.

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Triangular numbers

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1:

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+ x

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3:

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x x

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+ + x x

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6:

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x x

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x x x x

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+ + + x x x

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10:

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x x

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x x x x

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x x x x x x

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+ + + + x x x x

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Square numbers

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1:

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+ x

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4:

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x + x x

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+ + x x

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9:

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x x + x x x

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x x + x x x

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+ + + x x x

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16:

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x x x + x x x x

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x x x + x x x x

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x x x + x x x x

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+ + + + x x x x

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Polygons with higher numbers of sides, such as pentagons and hexagons, can also be represented as arrangements of dots (by convention 1 is the first polygonal number for any number of sides).

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Pentagonal numbers:

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1:

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+ x

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5:

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x x

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+ + x x

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+ + x x

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12:

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x x

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x x x x

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+ x x + x x x x

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+ + x x

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+ + + x x x

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22:

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x x

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x x x x

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x x x x x x x x

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+ x x + x x x x

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+ x x x + x x x x x

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+ + x x

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+ + + + x x x x

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35:

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x x

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x x x x

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x x x x x x x x

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x x x x x x x x

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+ x x x x x + x x x x x x x

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+ x x + x x x x

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+ x x x x + x x x x x x

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+ + x x

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+ + + + + x x x x x

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Hexagonal numbers

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1:

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x

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6:

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x x

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+ + x x

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+ + x x

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+ x

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15:

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x x

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x x x x

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+ x x + x x x x

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+ x + x x x

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+ + x x

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+ + x x

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+ x

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28:

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x x

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x x x x

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x x x x x x x x

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+ x x x + x x x x x

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+ x x + x x x x

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+ x x + x x x x

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+ x + x x x

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+ + x x

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+ + x x

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+ x

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45:

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x x

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x x x x

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x x x x x x x x

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x x x x x x x x x x

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+ x x x x + x x x x x x

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+ x x x x + x x x x x x

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+ x x x + x x x x x

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+ x x + x x x x

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+ x x + x x x x

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+ x + x x x

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+ + x x

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+ + x x

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+ x

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66: (which is also a triangular number and a sphenic number)

Related Topics:
Triangular number - Sphenic number

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x x

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x x x x

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x x x x x x x x

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x x x x x x x x x x

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x x x x x x x x x x x x

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+ x x x x x x + x x x x x x x x

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+ x x x x x + x x x x x x x

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+ x x x x + x x x x x x

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+ x x x x + x x x x x x

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+ x x x + x x x x x

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+ x x + x x x x

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+ x x + x x x x

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+ x + x x x

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+ + x x

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+ + x x

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+ x

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91:

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x x

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x x x x

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x x x x x x x x

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x x x x x x x x x x

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x x x x x x x x x x x x

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x x x x x x x x x x x x x x x x

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+ x x x x x x x + x x x x x x x x x

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+ x x x x x x + x x x x x x x x

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+ x x x x x x + x x x x x x x x

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+ x x x x x + x x x x x x x

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+ x x x x + x x x x x x

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+ x x x x + x x x x x x

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+ x x x + x x x x x

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+ x x + x x x x

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+ x x + x x x x

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+ x + x x x

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+ + x x

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+ + x x

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+ x

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If s is the number of sides in a polygon, the formula for the nth s-polygonal number is ½n((s-2)n - (s-4)).

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NameFormulan=12345678910111213

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Triangular½n(1n + 1)

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13610152128364555667891

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Square½n(2n - 0)

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149162536496481100121144169

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Pentagonal½n(3n - 1)

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15122235517092117145176210247

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Hexagonal½n(4n - 2)

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161528456691120153190231276325

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Heptagonal½n(5n - 3)

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1718345581112148189235286342403

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Octagonal½n(6n - 4)

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1821406596133176225280341408481

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Nonagonal½n(7n - 5)

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19244675111154204261325396474559

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Decagonal½n(8n - 6)

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110275285126175232297370451540637

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11-gonal½n(9n - 7)

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111305895141196260333415506606715

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12-gonal½n(10n - 8)

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1123364105156217288369460561672793

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13-gonal½n(11n - 9)

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1133670115171238316405505616738871

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14-gonal½n(12n - 10)

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1143976125186259344441550671804949

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15-gonal½n(13n - 11)

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11542821352012803724775957268701027

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16-gonal½n(14n - 12)

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11645881452163014005136407819361105

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17-gonal½n(15n - 13)

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117489415523132242854968583610021183

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18-gonal½n(16n - 14)

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1185110016524634345658573089110681261

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19-gonal½n(17n - 15)

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1195410617526136448462177594611341339

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20-gonal½n(18n - 16)

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12057112185276385512657820100112001417

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21-gonal½n(19n - 17)

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12160118195291406540693865105612661495

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22-gonal½n(20n - 18)

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12263124205306427568729910111113321573

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23-gonal½n(21n - 19)

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12366130215321448596765955116613981651

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24-gonal½n(22n - 20)

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124691362253364696248011000122114641729

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25-gonal½n(23n - 21)

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125721422353514906528371045127615301807

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26-gonal½n(24n - 22)

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126751482453665116808731090133115961885

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27-gonal½n(25n - 23)

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127781542553815327089091135138616621963

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28-gonal½n(26n - 24)

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128811602653965537369451180144117282041

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29-gonal½n(27n - 25)

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129841662754115747649811225149617942119

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30-gonal½n(28n - 26)

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1308717228542659579210171270155118602197

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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").

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