Polygonal number

In mathematics, a polygonal number is a number that counts dots arranged in the shape of a regular polygon. These are one type of 2-dimensional figurate numbers.

Polygonal numbers were first studied during the 6th century BC by the Ancient Greeks, who investigated and discussed properties of oblong, triangular, and square numbers.

Definition and examples

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

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

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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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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

The triangular number sequence is the representation of the numbers in the form of equilateral triangle arranged in a series or sequence. These numbers are in a sequence of 1, 3, 6, 10, 15, 21, 28, 36, 45, and so on.

Square numbers

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

Hexagonal numbers

Formula Polygonal numbers differ mainly by the difference between each consecutive number. The difference between each triangular number increases by 1 each iteration. For square numbers the difference increases by 2. For pentagonal numbers it increases by 3.

The size of the difference between each N starts at 1 for polygonal numbers (and for centered polygonal numbers the size of the difference starts at 0). For example: the first triangular is 1 and the "difference" starts at 1. To get the next triangular, you increment the difference by 1 and add it to the current one. The 2nd triangular number is 3, the difference is 2. Increment the difference again, it's 3. 3+3 is 6, the 3rd triangular. The difference between the 3rd and the 2nd is 3. Given that triangular numbers are the sum of the first N numbers this is unsurprising, however the same method holds for all other polygonal numbers as well.

With squares for the Nth and diff we start at the first square 1 and a diff (or offset) of 1. For the next square the difference is first incremented by 2: 1+2=3. We then add the difference to the current square: 1+3=4. The 2nd square number. Continuing the pattern for Nth+diff: 4+5=9, 9+7=16, 16+9=25. In other words, for the Nth polygonal number the difference between it and the next is X * N + 1. Where X is the number of sides (beginning at 1 for triangular).

If is the number of sides in a polygon, the formula for the th -gonal number is

The th -gonal number is also related to the triangular numbers as follows:

Thus:

For a given -gonal number , one can find by

and one can find by

Every hexagonal number is also a triangular number

Applying the formula above:

to the case of 6 sides gives:

but since:

it follows that:

This shows that the th hexagonal number is also the th triangular number . We can find every hexagonal number by simply taking the odd-numbered triangular numbers:

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, ...

Table of values The first six values in the column "sum of reciprocals", for triangular to octagonal numbers, come from a published solution to the general problem, which also gives a general formula for any number of sides, in terms of the digamma function. NameFormula Sum of reciprocals OEIS number 1 2 3 4 5 6 7 8 9 10 2 Natural (line segment) 1 2 3 4 5 6 7 8 9 10 ∞ (diverges) 3 Triangular 1 3 6 10 15 21 28 36 45 55 2 4 Square 1 4 9 16 25 36 49 64 81 100 5 Pentagonal 1 5 12 22 35 51 70 92 117 145 6 Hexagonal 1 6 15 28 45 66 91 120 153 190 7 Heptagonal 1 7 18 34 55 81 112 148 189 235 8 Octagonal 1 8 21 40 65 96 133 176 225 280 9 Nonagonal 1 9 24 46 75 111 154 204 261 325 10 Decagonal 1 10 27 52 85 126 175 232 297 370 11 Hendecagonal 1 11 30 58 95 141 196 260 333 415 12 Dodecagonal 1 12 33 64 105 156 217 288 369 460 13 Tridecagonal 1 13 36 70 115 171 238 316 405 505 14 Tetradecagonal 1 14 39 76 125 186 259 344 441 550 15 Pentadecagonal 1 15 42 82 135 201 280 372 477 595 16 Hexadecagonal 1 16 45 88 145 216 301 400 513 640 17 Heptadecagonal 1 17 48 94 155 231 322 428 549 685 18 Octadecagonal 1 18 51 100 165 246 343 456 585 730 19 Enneadecagonal 1 19 54 106 175 261 364 484 621 775 20 Icosagonal 1 20 57 112 185 276 385 512 657 820 21 Icosihenagonal 1 21 60 118 195 291 406 540 693 865 22 Icosidigonal 1 22 63 124 205 306 427 568 729 910 23 Icositrigonal 1 23 66 130 215 321 448 596 765 955 24 Icositetragonal 1 24 69 136 225 336 469 624 801 1000 =

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").

A property of this table can be expressed by the following identity (see ):

with

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 numbers.

The following table summarizes the set of -gonal -gonal numbers for small values of and . {| class="wikitable" border="1" |- ! ! ! Sequence ! OEIS number |- |4 |3 |1, 36, 1225, 41616, 1413721, 48024900, 1631432881, 55420693056, 1882672131025, 63955431761796, 2172602007770041, 73804512832419600, 2507180834294496361, 85170343853180456676, 2893284510173841030625, 98286503002057414584576, 3338847817559778254844961, ...

|5 |3 |1, 210, 40755, 7906276, 1533776805, 297544793910, 57722156241751, 11197800766105800, 2172315626468283465, …

|5 |4 |1, 9801, 94109401, 903638458801, 8676736387298001, 83314021887196947001, 799981229484128697805801, ...

|6 |3 |All hexagonal numbers are also triangular.

|6 |4 |1, 1225, 1413721, 1631432881, 1882672131025, 2172602007770041, 2507180834294496361, 2893284510173841030625, 3338847817559778254844961, 3853027488179473932250054441, ...

|6 |5 |1, 40755, 1533776805, …

|7 |3 |1, 55, 121771, 5720653, 12625478965, 593128762435, 1309034909945503, 61496776341083161, 135723357520344181225, 6376108764003055554511, 14072069153115290487843091, …

|7 |4 |1, 81, 5929, 2307361, 168662169, 12328771225, 4797839017609, 350709705290025, 25635978392186449, 9976444135331412025, …

|7 |5 |1, 4347, 16701685, 64167869935, …

|7 |6 |1, 121771, 12625478965, …

|8 |3 |1, 21, 11781, 203841, …

|8 |4 |1, 225, 43681, 8473921, 1643897025, 318907548961, 61866420601441, 12001766689130625, 2328280871270739841, 451674487259834398561, 87622522247536602581025, 16998317641534841066320321, …

|8 |5 |1, 176, 1575425, 234631320, …

|8 |6 |1, 11781, 113123361, …

|8 |7 |1, 297045, 69010153345, …

|9 |3 |1, 325, 82621, 20985481, …

|9 |4 |1, 9, 1089, 8281, 978121, 7436529, 878351769, 6677994961, 788758910641, 5996832038649, 708304623404049, 5385148492712041, 636056763057925561, ...

|9 |5 |1, 651, 180868051, …

|9 |6 |1, 325, 5330229625, …

|9 |7 |1, 26884, 542041975, …

|9 |8 |1, 631125, 286703855361, …

In some cases, such as and , there are no numbers in both sets other than 1.

The problem of finding numbers that belong to three polygonal sets is more difficult. Katayama proved that if three different integers , , and are all at least 3 and not equal to 6, then only finitely many numbers are simultaneously -gonal, -gonal, and -gonal.

Katayama, Furuya, and Nishioka proved that if the integer is such that or , then the only -gonal square triangular number is 1. For example, that paper gave the following proof for the case where .Ibid., p. 4. Suppose that for some positive integers , , and . A calculation shows that the point defined by is on the curve . That fact forces (as an elliptic curve database confirms), so and the result follows.

The number 1225 is hecatonicositetragonal (), hexacontagonal (), icosienneagonal (), hexagonal, square, and triangular.

Polygonal number

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