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Polygonal number

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Template:Short description In mathematics, a polygonal number is a number that counts dots arranged in the shape of a regular polygonTemplate:R. 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 numbersTemplate:R.

Definition and examplesEdit

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 numbersEdit

File:Polygonal Number 3.gif

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 numbersEdit

File:Polygonal Number 4.gif

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 numbersEdit

File:Polygonal Number 5.gif

Hexagonal numbersEdit

File:Polygonal Number 6.gif

FormulaEdit

File:Visual proof polygonal numbers.svg
An s-gonal number greater than 1 can be decomposed into s−2 triangular numbers and a natural number.

If Template:Mvar is the number of sides in a polygon, the formula for the Template:Mvarth Template:Mvar-gonal number Template:Math is

<math>P(s,n) = \frac{(s-2)n^2-(s-4)n}{2}</math>

The Template:Mvarth Template:Mvar-gonal number is also related to the triangular numbers Template:Math as follows:<ref name=":0">Template:Cite book</ref>

<math>P(s,n) = (s-2)T_{n-1} + n = (s-3)T_{n-1} + T_n\, .</math>

Thus:

<math>\begin{align}

P(s,n+1)-P(s,n) &= (s-2)n + 1\, ,\\ P(s+1,n) - P(s,n) &= T_{n-1} = \frac{n(n-1)}{2}\, ,\\ P(s+k,n) - P(s,n) &= k T_{n-1} = k\frac{n(n-1)}{2}\, . \end{align}</math>

For a given Template:Mvar-gonal number Template:Math, one can find Template:Mvar by

<math>n = \frac{\sqrt{8(s-2)x+{(s-4)}^2}+(s-4)}{2(s-2)}</math>

and one can find Template:Mvar by

<math>s = 2+\frac{2}{n}\cdot\frac{x-n}{n-1}</math>.

Every hexagonal number is also a triangular numberEdit

Template:CSS image crop Applying the formula above:

<math>P(s,n) = (s-2)T_{n-1} + n </math>

to the case of 6 sides gives:

<math>P(6,n) = 4T_{n-1} + n </math>

but since:

<math>T_{n-1} = \frac{n(n-1)}{2} </math>

it follows that:

<math>P(6,n) = \frac{4n(n-1)}{2} + n = \frac{2n(2n-1)}{2} = T_{2n-1}</math>

This shows that the Template:Mvarth hexagonal number Template:Math is also the Template:Mathth triangular number Template:Math. We can find every hexagonal number by simply taking the odd-numbered triangular numbers:<ref name=":0" />

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

Table of valuesEdit

The first 6 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.<ref name="siam_07-003s">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Template:Mvar Name Formula Template:Mvar Sum of reciprocals<ref name="siam_07-003s" /><ref>{{#invoke:citation/CS1|citation CitationClass=web

}}</ref>

OEIS number
1 2 3 4 5 6 7 8 9 10
2 Natural (line segment) Template:Math 1 2 3 4 5 6 7 8 9 10 ∞ (diverges) A000027
3 Triangular Template:Math 1 3 6 10 15 21 28 36 45 55 2<ref name="siam_07-003s" /> A000217
4 Square Template:Math 1 4 9 16 25 36 49 64 81 100 Template:Sfrac<ref name="siam_07-003s" /> A000290
5 Pentagonal Template:Math 1 5 12 22 35 51 70 92 117 145 Template:Math<ref name="siam_07-003s" /> A000326
6 Hexagonal Template:Math 1 6 15 28 45 66 91 120 153 190 Template:Math<ref name="siam_07-003s" /> A000384
7 Heptagonal Template:Math 1 7 18 34 55 81 112 148 189 235 <math>\begin{matrix}

\tfrac{2}{3}\ln 5 \\ +\tfrac{{1}+\sqrt{5}}{3}\ln\tfrac\sqrt{10-2\sqrt{5}}{2} \\ +\tfrac{{1}-\sqrt{5}}{3}\ln\tfrac\sqrt{10+2\sqrt{5}}{2} \\ +\tfrac{\pi\sqrt{25-10\sqrt{5}}}{15} \end{matrix}</math><ref name="siam_07-003s" />

A000566
8 Octagonal Template:Math 1 8 21 40 65 96 133 176 225 280 Template:Math<ref name="siam_07-003s" /> A000567
9 Nonagonal Template:Math 1 9 24 46 75 111 154 204 261 325 A001106
10 Decagonal Template:Math 1 10 27 52 85 126 175 232 297 370 Template:Math A001107
11 Hendecagonal Template:Math 1 11 30 58 95 141 196 260 333 415 A051682
12 Dodecagonal Template:Math 1 12 33 64 105 156 217 288 369 460 A051624
13 Tridecagonal Template:Math 1 13 36 70 115 171 238 316 405 505 A051865
14 Tetradecagonal Template:Math 1 14 39 76 125 186 259 344 441 550 Template:Math A051866
15 Pentadecagonal Template:Math 1 15 42 82 135 201 280 372 477 595 A051867
16 Hexadecagonal Template:Math 1 16 45 88 145 216 301 400 513 640 A051868
17 Heptadecagonal Template:Math 1 17 48 94 155 231 322 428 549 685 A051869
18 Octadecagonal Template:Math 1 18 51 100 165 246 343 456 585 730 Template:Math Template:Math A051870
19 Enneadecagonal Template:Math 1 19 54 106 175 261 364 484 621 775 A051871
20 Icosagonal Template:Math 1 20 57 112 185 276 385 512 657 820 A051872
21 Icosihenagonal Template:Math 1 21 60 118 195 291 406 540 693 865 A051873
22 Icosidigonal Template:Math 1 22 63 124 205 306 427 568 729 910 A051874
23 Icositrigonal Template:Math 1 23 66 130 215 321 448 596 765 955 A051875
24 Icositetragonal Template:Math 1 24 69 136 225 336 469 624 801 1000 A051876
... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
10000 Myriagonal Template:Math 1 10000 29997 59992 99985 149976 209965 279952 359937 449920 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").

A property of this table can be expressed by the following identity (see A086270):

<math>2\,P(s,n) = P(s+k,n) + P(s-k,n),</math>

with

<math>k = 0, 1, 2, 3, ..., s-3.</math>

CombinationsEdit

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 Template:Mvar-gonal Template:Mvar-gonal numbers for small values of Template:Mvar and Template:Mvar.

Template:Mvar Template:Mvar Sequence OEIS number
4 3 1, 36, 1225, 41616, 1413721, 48024900, 1631432881, 55420693056, 1882672131025, 63955431761796, 2172602007770041, 73804512832419600, 2507180834294496361, 85170343853180456676, 2893284510173841030625, 98286503002057414584576, 3338847817559778254844961, ... A001110
5 3 1, 210, 40755, 7906276, 1533776805, 297544793910, 57722156241751, 11197800766105800, 2172315626468283465, … A014979
5 4 1, 9801, 94109401, 903638458801, 8676736387298001, 83314021887196947001, 799981229484128697805801, ... A036353
6 3 All hexagonal numbers are also triangular. A000384
6 4 1, 1225, 1413721, 1631432881, 1882672131025, 2172602007770041, 2507180834294496361, 2893284510173841030625, 3338847817559778254844961, 3853027488179473932250054441, ... A046177
6 5 1, 40755, 1533776805, … A046180
7 3 1, 55, 121771, 5720653, 12625478965, 593128762435, 1309034909945503, 61496776341083161, 135723357520344181225, 6376108764003055554511, 14072069153115290487843091, … A046194
7 4 1, 81, 5929, 2307361, 168662169, 12328771225, 4797839017609, 350709705290025, 25635978392186449, 9976444135331412025, … A036354
7 5 1, 4347, 16701685, 64167869935, … A048900
7 6 1, 121771, 12625478965, … A048903
8 3 1, 21, 11781, 203841, … A046183
8 4 1, 225, 43681, 8473921, 1643897025, 318907548961, 61866420601441, 12001766689130625, 2328280871270739841, 451674487259834398561, 87622522247536602581025, 16998317641534841066320321, … A036428
8 5 1, 176, 1575425, 234631320, … A046189
8 6 1, 11781, 113123361, … A046192
8 7 1, 297045, 69010153345, … A048906
9 3 1, 325, 82621, 20985481, … A048909
9 4 1, 9, 1089, 8281, 978121, 7436529, 878351769, 6677994961, 788758910641, 5996832038649, 708304623404049, 5385148492712041, 636056763057925561, ... A036411
9 5 1, 651, 180868051, … A048915
9 6 1, 325, 5330229625, … A048918
9 7 1, 26884, 542041975, … A048921
9 8 1, 631125, 286703855361, … A048924

In some cases, such as Template:Math and Template:Math, 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 other such numbers has yet to be found.<ref>{{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web |_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:PentagonalSquareTriangularNumber%7CPentagonalSquareTriangularNumber.html}} |title = Pentagonal Square Triangular Number |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}</ref>

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

See alsoEdit

NotesEdit

Template:Reflist

ReferencesEdit

|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:PolygonalNumber%7CPolygonalNumber.html}} |title = Polygonal Numbers |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}

External linksEdit

Template:Classes of natural numbers Template:Series (mathematics) Template:Authority control

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