Polygonal number

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 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

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 numbers

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 numbers

File:Polygonal Number 5.gif

Hexagonal numbers

File:Polygonal Number 6.gif

Formula

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

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

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:

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

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 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.

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:Calculator	∞ (diverges)	A000027
3	Triangular	Template:Math	1	3	6	10	15	21	28	36	45	55	Template:Calculator	2	A000217
4	Square	Template:Math	1	4	9	16	25	36	49	64	81	100	Template:Calculator	Template:Sfrac	A000290
5	Pentagonal	Template:Math	1	5	12	22	35	51	70	92	117	145	Template:Calculator	Template:Math	A000326
6	Hexagonal	Template:Math	1	6	15	28	45	66	91	120	153	190	Template:Calculator	Template:Math	A000384
7	Heptagonal	Template:Math	1	7	18	34	55	81	112	148	189	235	Template:Calculator	$\begin{array}{c}\frac{2}{3}\ln 5\\ +\frac{1+\sqrt{5}}{3}\ln \frac{\sqrt{10-2\sqrt{5}}}{2}\\ +\frac{1-\sqrt{5}}{3}\ln \frac{\sqrt{10+2\sqrt{5}}}{2}\\ +\frac{\pi \sqrt{25-10\sqrt{5}}}{15}\end{array}$	A000566
8	Octagonal	Template:Math	1	8	21	40	65	96	133	176	225	280	Template:Calculator	Template:Math	A000567
9	Nonagonal	Template:Math	1	9	24	46	75	111	154	204	261	325	Template:Calculator		A001106
10	Decagonal	Template:Math	1	10	27	52	85	126	175	232	297	370	Template:Calculator	Template:Math	A001107
11	Hendecagonal	Template:Math	1	11	30	58	95	141	196	260	333	415	Template:Calculator		A051682
12	Dodecagonal	Template:Math	1	12	33	64	105	156	217	288	369	460	Template:Calculator		A051624
13	Tridecagonal	Template:Math	1	13	36	70	115	171	238	316	405	505	Template:Calculator		A051865
14	Tetradecagonal	Template:Math	1	14	39	76	125	186	259	344	441	550	Template:Calculator	Template:Math	A051866
15	Pentadecagonal	Template:Math	1	15	42	82	135	201	280	372	477	595	Template:Calculator		A051867
16	Hexadecagonal	Template:Math	1	16	45	88	145	216	301	400	513	640	Template:Calculator		A051868
17	Heptadecagonal	Template:Math	1	17	48	94	155	231	322	428	549	685	Template:Calculator		A051869
18	Octadecagonal	Template:Math	1	18	51	100	165	246	343	456	585	730	Template:Calculator	Template:Math Template:Math	A051870
19	Enneadecagonal	Template:Math	1	19	54	106	175	261	364	484	621	775	Template:Calculator		A051871
20	Icosagonal	Template:Math	1	20	57	112	185	276	385	512	657	820	Template:Calculator		A051872
21	Icosihenagonal	Template:Math	1	21	60	118	195	291	406	540	693	865	Template:Calculator		A051873
22	Icosidigonal	Template:Math	1	22	63	124	205	306	427	568	729	910	Template:Calculator		A051874
23	Icositrigonal	Template:Math	1	23	66	130	215	321	448	596	765	955	Template:Calculator		A051875
24	Icositetragonal	Template:Math	1	24	69	136	225	336	469	624	801	1000	Template:Calculator		A051876
Template:Calculator label = Template:Calculator		Template:Math	Template:Calculator	Template:Calculator	Template:Calculator	Template:Calculator	Template:Calculator	Template:Calculator	Template:Calculator	Template:Calculator	Template:Calculator	Template:Calculator	Template:Calculator

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):

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

with

$k=0,1,2,3,...,s-3.$

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, ...	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.

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

References

• The Penguin Dictionary of Curious and Interesting Numbers, David Wells (Penguin Books, 1997) [[[:Template:Isbn]]].
• Polygonal numbers at PlanetMath

External links

• Template:Springer
• Polygonal Numbers: Every s-polygonal number between 1 and 1000 clickable for 2<=s<=337
• Template:Replace on YouTube
• Polygonal Number Counting Function: http://www.mathisfunforum.com/viewtopic.php?id=17853