Polygonal number

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

Definition and examples

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

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

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

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).

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

$P(s,n)=\frac{(s-2)n^2-(s-4)n}{2}$

The Template:Mvarth Template:Mvar-gonal number is also related to the triangular numbers Template:Math as follows:

$P(s,n)=(s-2)T_{n-1}+n=(s-3)T_{n-1}+T_n.$

Thus:

$\begin{array}{cc}P(s,n+1)-P(s,n)& =(s-2)n+1,\\ P(s+1,n)-P(s,n)& =T_{n-1}=\frac{n(n-1)}{2},\\ P(s+k,n)-P(s,n)& =k{T}_{n-1}=k\frac{n(n-1)}{2}.\end{array}$

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

$n=\frac{\sqrt{8(s-2)x+{(s-4)}^2}+(s-4)}{2(s-2)}$

and one can find Template:Mvar by

$s=2+\frac{2}{n}\cdot \frac{x-n}{n-1}$.

Every hexagonal number is also a triangular number

Template:CSS image crop Applying the formula above:

$P(s,n)=(s-2)T_{n-1}+n$

to the case of 6 sides gives:

$P(6,n)=4T_{n-1}+n$

but since:

$T_{n-1}=\frac{n(n-1)}{2}$

it follows that:

$P(6,n)=\frac{4n(n-1)}{2}+n=\frac{2n(2n-1)}{2}=T_{2n-1}$

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:

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.

Template:Mvar	Name	Formula	Template:Mvar											Sum of reciprocals	OEIS number
			1	2	3	4	5	6	7	8	9	10	Template:Calculator
2	Natural (line segment)	Template:Math	1	2	3	4	5	6	7	8	9	10	Template:Math	∞ (diverges)	Template:OEIS link
3	Triangular	Template:Math	1	3	6	10	15	21	28	36	45	55	Template:Math	2	Template:OEIS link
4	Square	Template:Math	1	4	9	16	25	36	49	64	81	100	Template:Math	Template:SfracTemplate:Efn-lg	Template:OEIS link
5	Pentagonal	Template:Math	1	5	12	22	35	51	70	92	117	145	Template:Math	Template:Math	Template:OEIS link
6	Hexagonal	Template:Math	1	6	15	28	45	66	91	120	153	190	Template:Math	Template:Math	Template:OEIS link
7	Heptagonal	Template:Math	1	7	18	34	55	81	112	148	189	235	Template:Math	$\begin{array}{c}\frac{2}{3}\ln 5\\ +\frac{1+\sqrt{5}}{3}\ln \frac{\sqrt{10-2\sqrt{5}}}{2}\\ +\frac{1-\sqrt{5}}{3}\ln \frac{\sqrt{10+2\sqrt{5}}}{2}\\ +\frac{\pi \sqrt{25-10\sqrt{5}}}{15}\end{array}$	Template:OEIS link
8	Octagonal	Template:Math	1	8	21	40	65	96	133	176	225	280	Template:Math	Template:Math	Template:OEIS link
9	Nonagonal	Template:Math	1	9	24	46	75	111	154	204	261	325	Template:Math		Template:OEIS link
10	Decagonal	Template:Math	1	10	27	52	85	126	175	232	297	370	Template:Math	Template:Math	Template:OEIS link
11	Hendecagonal	Template:Math	1	11	30	58	95	141	196	260	333	415	Template:Math		Template:OEIS link
12	Dodecagonal	Template:Math	1	12	33	64	105	156	217	288	369	460	Template:Math		Template:OEIS link
13	Tridecagonal	Template:Math	1	13	36	70	115	171	238	316	405	505	Template:Math		Template:OEIS link
14	Tetradecagonal	Template:Math	1	14	39	76	125	186	259	344	441	550	Template:Math	Template:Math	Template:OEIS link
15	Pentadecagonal	Template:Math	1	15	42	82	135	201	280	372	477	595	Template:Math		Template:OEIS link
16	Hexadecagonal	Template:Math	1	16	45	88	145	216	301	400	513	640	Template:Math		Template:OEIS link
17	Heptadecagonal	Template:Math	1	17	48	94	155	231	322	428	549	685	Template:Math		Template:OEIS link
18	Octadecagonal	Template:Math	1	18	51	100	165	246	343	456	585	730	Template:Math	Template:Math Template:Math	Template:OEIS link
19	Enneadecagonal	Template:Math	1	19	54	106	175	261	364	484	621	775	Template:Math		Template:OEIS link
20	Icosagonal	Template:Math	1	20	57	112	185	276	385	512	657	820	Template:Math		Template:OEIS link
21	Icosihenagonal	Template:Math	1	21	60	118	195	291	406	540	693	865	Template:Math		Template:OEIS link
22	Icosidigonal	Template:Math	1	22	63	124	205	306	427	568	729	910	Template:Math		Template:OEIS link
23	Icositrigonal	Template:Math	1	23	66	130	215	321	448	596	765	955	Template:Math		Template:OEIS link
24	Icositetragonal	Template:Math	1	24	69	136	225	336	469	624	801	1000	Template:Math		Template:OEIS link
Template:Calculator label = Template:Calculator		Template:Math	Template:Math	Template:Math	Template:Math	Template:Math	Template:Math	Template:Math	Template:Math	Template:Math	Template:Math	Template:Math	Template:Math

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 Template:OEIS link):

$2P(s,n)=P(s+k,n)+P(s-k,n),$

with

$k=0,1,2,3,\mathrm{...},s-3.$

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 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, ...	Template:OEIS link
5	3	1, 210, 40755, 7906276, 1533776805, 297544793910, 57722156241751, 11197800766105800, 2172315626468283465, …	Template:OEIS link
5	4	1, 9801, 94109401, 903638458801, 8676736387298001, 83314021887196947001, 799981229484128697805801, ...	Template:OEIS link
6	3	All hexagonal numbers are also triangular.	Template:OEIS link
6	4	1, 1225, 1413721, 1631432881, 1882672131025, 2172602007770041, 2507180834294496361, 2893284510173841030625, 3338847817559778254844961, 3853027488179473932250054441, ...	Template:OEIS link
6	5	1, 40755, 1533776805, …	Template:OEIS link
7	3	1, 55, 121771, 5720653, 12625478965, 593128762435, 1309034909945503, 61496776341083161, 135723357520344181225, 6376108764003055554511, 14072069153115290487843091, …	Template:OEIS link
7	4	1, 81, 5929, 2307361, 168662169, 12328771225, 4797839017609, 350709705290025, 25635978392186449, 9976444135331412025, …	Template:OEIS link
7	5	1, 4347, 16701685, 64167869935, …	Template:OEIS link
7	6	1, 121771, 12625478965, …	Template:OEIS link
8	3	1, 21, 11781, 203841, …	Template:OEIS link
8	4	1, 225, 43681, 8473921, 1643897025, 318907548961, 61866420601441, 12001766689130625, 2328280871270739841, 451674487259834398561, 87622522247536602581025, 16998317641534841066320321, …	Template:OEIS link
8	5	1, 176, 1575425, 234631320, …	Template:OEIS link
8	6	1, 11781, 113123361, …	Template:OEIS link
8	7	1, 297045, 69010153345, …	Template:OEIS link
9	3	1, 325, 82621, 20985481, …	Template:OEIS link
9	4	1, 9, 1089, 8281, 978121, 7436529, 878351769, 6677994961, 788758910641, 5996832038649, 708304623404049, 5385148492712041, 636056763057925561, ...	Template:OEIS link
9	5	1, 651, 180868051, …	Template:OEIS link
9	6	1, 325, 5330229625, …	Template:OEIS link
9	7	1, 26884, 542041975, …	Template:OEIS link
9	8	1, 631125, 286703855361, …	Template:OEIS link

In some cases, such as Template:Math and Template:Math, there are no numbers in both sets other than 1.Template:Citation needed

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

Katayama, Furuya, and Nishioka proved that if the integer Template:Mvar is such that $s=5$ or $7\le s\le 12$, then the only Template:Mvar-gonal square triangular number is 1. For example, that paper gave the following proof for the case where $s=5$. Suppose that $P(3,n)=P(4,p)=P(5,q)$ for some positive integers Template:Mvar, Template:Mvar, and Template:Mvar. A calculation shows that the point $(x,y)$ defined by $(x,y)=(48p^2+3,24p(2n+1)(6q-1))$ is on the curve $Y^2=X^3-X^2-9X+9$. That fact forces $(x,y)=(51,360)$ (as an elliptic curve database confirms), so $p=1$ and the result follows.

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

See also

• Centered polygonal number
• Polyhedral number
• Fermat polygonal number theorem

Bibliography

• The Penguin Dictionary of Curious and Interesting Numbers, David Wells (Penguin Books, 1997) [[[:Template:Isbn]]].
• Template:PlanetMath
• Template:MathWorld
• Template:Cite book

External links

• Template:Springer
• Polygonal Numbers: Every s-polygonal number between 1 and 1000 clickable for 2<=s<=337
• Template:YouTube
• Polygonal Number Counting Function