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

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 $P (s, n)$ Script error: No such module "Check for unknown parameters". 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 $T n$ Script error: No such module "Check for unknown parameters". as follows:^[1]

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

Thus:

\begin{aligned} 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{aligned}

For a given Template:Mvar-gonal number $P (s, n) = x$ Script error: No such module "Check for unknown parameters"., 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) = 4 T_{n - 1} + n

but since:

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

it follows that:

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

This shows that the Template:Mvarth hexagonal number $P (6, n)$ Script error: No such module "Check for unknown parameters". is also the $(2 n - 1)$ Script error: No such module "Check for unknown parameters".th triangular number $T 2 n -1$ Script error: No such module "Check for unknown parameters".. We can find every hexagonal number by simply taking the odd-numbered triangular numbers:^[1]

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.^[2]

Template:Mvar	Name	Formula	Template:Mvar											Sum of reciprocals^[2]^[3]	OEIS number
Template:Mvar	Name	Formula	1	2	3	4	5	6	7	8	9	10	Template:Calculator	Sum of reciprocals^[2]^[3]	OEIS number
2	Natural (line segment)	$Template:Sfrac(0n2 + 2n) = n$ Script error: No such module "Check for unknown parameters".	1	2	3	4	5	6	7	8	9	10	Template:Calculator	∞ (diverges)	A000027
3	Triangular	$Template:Sfrac(n2 + n)$ Script error: No such module "Check for unknown parameters".	1	3	6	10	15	21	28	36	45	55	Template:Calculator	2^[2]	A000217
4	Square	$Template:Sfrac(2n2 − 0n) = n2$ Script error: No such module "Check for unknown parameters".	1	4	9	16	25	36	49	64	81	100	Template:Calculator	Template:Sfrac^[2]Template:Efn-lg	A000290
5	Pentagonal	$Template:Sfrac(3n2 − n)$ Script error: No such module "Check for unknown parameters".	1	5	12	22	35	51	70	92	117	145	Template:Calculator	$3 ln 3 − Template:Sfrac$ Script error: No such module "Check for unknown parameters".^[2]	A000326
6	Hexagonal	$Template:Sfrac(4n2 − 2n) = 2n2 − n$ Script error: No such module "Check for unknown parameters".	1	6	15	28	45	66	91	120	153	190	Template:Calculator	$2 ln 2$ Script error: No such module "Check for unknown parameters".^[2]	A000384
7	Heptagonal	$Template:Sfrac(5n2 − 3n)$ Script error: No such module "Check for unknown parameters".	1	7	18	34	55	81	112	148	189	235	Template:Calculator	$\begin{matrix} \frac{2}{3} \ln 5 \\ + \frac{1 + \sqrt{5}}{3} \ln \frac{\sqrt{10 - 2 \sqrt{5}}}{2} \\ + \frac{1 - \sqrt{5}}{3} \ln \frac{\sqrt{10 + 2 \sqrt{5}}}{2} \\ + \frac{π \sqrt{25 - 10 \sqrt{5}}}{15} \end{matrix}$ ^[2]	A000566
8	Octagonal	$Template:Sfrac(6n2 − 4n) = 3n2 − 2n$ Script error: No such module "Check for unknown parameters".	1	8	21	40	65	96	133	176	225	280	Template:Calculator	$Template:Sfrac ln 3 + Template:Sfrac$ Script error: No such module "Check for unknown parameters".^[2]	A000567
9	Nonagonal	$Template:Sfrac(7n2 − 5n)$ Script error: No such module "Check for unknown parameters".	1	9	24	46	75	111	154	204	261	325	Template:Calculator		A001106
10	Decagonal	$Template:Sfrac(8n2 − 6n) = 4n2 − 3n$ Script error: No such module "Check for unknown parameters".	1	10	27	52	85	126	175	232	297	370	Template:Calculator	$ln 2 + Template:Sfrac$ Script error: No such module "Check for unknown parameters".	A001107
11	Hendecagonal	$Template:Sfrac(9n2 − 7n)$ Script error: No such module "Check for unknown parameters".	1	11	30	58	95	141	196	260	333	415	Template:Calculator		A051682
12	Dodecagonal	$Template:Sfrac(10n2 − 8n)$ Script error: No such module "Check for unknown parameters".	1	12	33	64	105	156	217	288	369	460	Template:Calculator		A051624
13	Tridecagonal	$Template:Sfrac(11n2 − 9n)$ Script error: No such module "Check for unknown parameters".	1	13	36	70	115	171	238	316	405	505	Template:Calculator		A051865
14	Tetradecagonal	$Template:Sfrac(12n2 − 10n)$ Script error: No such module "Check for unknown parameters".	1	14	39	76	125	186	259	344	441	550	Template:Calculator	$Template:Sfrac ln 2 + Template:Sfrac ln 3 + Template:Sfrac$ Script error: No such module "Check for unknown parameters".	A051866
15	Pentadecagonal	$Template:Sfrac(13n2 − 11n)$ Script error: No such module "Check for unknown parameters".	1	15	42	82	135	201	280	372	477	595	Template:Calculator		A051867
16	Hexadecagonal	$Template:Sfrac(14n2 − 12n)$ Script error: No such module "Check for unknown parameters".	1	16	45	88	145	216	301	400	513	640	Template:Calculator		A051868
17	Heptadecagonal	$Template:Sfrac(15n2 − 13n)$ Script error: No such module "Check for unknown parameters".	1	17	48	94	155	231	322	428	549	685	Template:Calculator		A051869
18	Octadecagonal	$Template:Sfrac(16n2 − 14n)$ Script error: No such module "Check for unknown parameters".	1	18	51	100	165	246	343	456	585	730	Template:Calculator	$Template:Sfrac ln 2 − Template:Sfrac ln (3 − 2 REDIRECT Template:Radic Template:Rcat shell)Script error: No such module "Check for unknown parameters". + Template:SfracScript error: No such module "Check for unknown parameters".$	A051870
19	Enneadecagonal	$Template:Sfrac(17n2 − 15n)$ Script error: No such module "Check for unknown parameters".	1	19	54	106	175	261	364	484	621	775	Template:Calculator		A051871
20	Icosagonal	$Template:Sfrac(18n2 − 16n)$ Script error: No such module "Check for unknown parameters".	1	20	57	112	185	276	385	512	657	820	Template:Calculator		A051872
21	Icosihenagonal	$Template:Sfrac(19n2 − 17n)$ Script error: No such module "Check for unknown parameters".	1	21	60	118	195	291	406	540	693	865	Template:Calculator		A051873
22	Icosidigonal	$Template:Sfrac(20n2 − 18n)$ Script error: No such module "Check for unknown parameters".	1	22	63	124	205	306	427	568	729	910	Template:Calculator		A051874
23	Icositrigonal	$Template:Sfrac(21n2 − 19n)$ Script error: No such module "Check for unknown parameters".	1	23	66	130	215	321	448	596	765	955	Template:Calculator		A051875
24	Icositetragonal	$Template:Sfrac(22n2 − 20n)$ Script error: No such module "Check for unknown parameters".	1	24	69	136	225	336	469	624	801	1000	Template:Calculator		A051876
Template:Calculator label = Template:Calculator		$Template:Sfrac(Template:Calculatorn2 − Template:Calculatorn)$ Script error: No such module "Check for unknown parameters".	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):

2 P (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 $s = 10$ Script error: No such module "Check for unknown parameters". and $t = 4$ Script error: No such module "Check for unknown parameters"., there are no numbers in both sets other than 1.Script error: No such module "Unsubst".

The problem of finding numbers that belong to three polygonal sets is more difficult. Katayama^[4] 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^[5] proved that if the integer Template:Mvar is such that $s = 5$ or $7 \leq s \leq 12$ , then the only Template:Mvar-gonal square triangular number is 1. For example, that paper gave the following proof for the case where $s = 5$ .^[6] 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 $(x, y) = (48 p^{2} + 3, 24 p (2 n + 1) (6 q - 1))$ is on the curve $Y^{2} = X^{3} - X^{2} - 9 X + 9$ . That fact forces $(x, y) = (51, 360)$ (as an elliptic curve database^[7] confirms), so $p = 1$ and the result follows.

The number 1225 is hecatonicositetragonal ( $s = 124$ Script error: No such module "Check for unknown parameters".), hexacontagonal ( $s = 60$ Script error: No such module "Check for unknown parameters".), icosienneagonal ( $s = 29$ Script error: No such module "Check for unknown parameters".), hexagonal, square, and triangular.

Notes

Script error: No such module "Check for unknown parameters".

References

↑ ^a ^b Script error: No such module "citation/CS1".
↑ ^a ^b ^c ^d ^e ^f ^g ^h Script error: No such module "citation/CS1".
↑ Script error: No such module "citation/CS1".
↑ Script error: No such module "Citation/CS1".
↑ Script error: No such module "Citation/CS1".
↑ Ibid., p. 4.
↑ Script error: No such module "citation/CS1".

Cite error: <ref> tag with name "tattersall2005" defined in <references> is not used in prior text.

Script error: No such module "Check for unknown parameters".

Bibliography

The Penguin Dictionary of Curious and Interesting Numbers, David Wells (Penguin Books, 1997) [[[:Template:Isbn]]].
Polygonal number at PlanetMath.
Script error: No such module "Template wrapper".
Script error: No such module "citation/CS1".

External links

Template:Springer
Polygonal Numbers: Every s-polygonal number between 1 and 1000 clickable for 2<=s<=337
Template:Replace on YouTubeScript error: No such module "Check for unknown parameters".
Polygonal Number Counting Function

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

[:0-1] Script error: No such module "citation/CS1".

[siam_07-003s-2] ↑ ^a ^b ^c ^d ^e ^f ^g ^h Script error: No such module "citation/CS1".

[3] Script error: No such module "citation/CS1".

[4] Script error: No such module "Citation/CS1".

[5] Script error: No such module "Citation/CS1".

[6] Ibid., p. 4.

[7] Script error: No such module "citation/CS1".

[1]

[2]

[3]

[4]

[5]

[6]

[7]

Polygonal number

Contents

Definition and examples

Triangular numbers

Square numbers

Pentagonal numbers

Hexagonal numbers

Formula

Every hexagonal number is also a triangular number

Table of values

Combinations

See also

Notes

References

Bibliography

External links

Navigation menu

Polygonal number

Definition and examples

Triangular numbers

Square numbers

Pentagonal numbers

Hexagonal numbers

Formula

Every hexagonal number is also a triangular number

Table of values

Combinations

See also

Notes

References

Bibliography

External links

Navigation menu

Search