Polygonal number

Last updated May 23, 2024

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.

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

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

If $s$ is the number of sides in a polygon, the formula for the $n$ th $s$ -gonal number $P (s, n)$ is

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

or

P(s,n)=(s-2){\frac {n(n-1)}{2}}+n

The $n$ th $s$ -gonal number is also related to the triangular numbers $T n$ 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)&=kT_{n-1}=k{\frac {n(n-1)}{2}}\,.\end{aligned}}

For a given $s$ -gonal number $P (s, n) = x$ , one can find $n$ by

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

and one can find $s$ by

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

.

Every hexagonal number is also a triangular number

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 $n$ th hexagonal number $P (6, n)$ is also the $(2 n - 1)$ th triangular number $T 2 n -1$ . 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 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.^[2]

$s$	Name	Formula	$n$										Sum of reciprocals^[2]^[3]	OEIS number
$s$	Name	Formula	1	2	3	4	5	6	7	8	9	10	Sum of reciprocals^[2]^[3]	OEIS number
2	Natural (line segment)	$.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}1/2(0n2 + 2n) = n$	1	2	3	4	5	6	7	8	9	10	∞ (diverges)	A000027
3	Triangular	$1 / 2 (n 2 + n)$	1	3	6	10	15	21	28	36	45	55	2^[2]	A000217
4	Square	$1 / 2 (2 n 2 - 0 n) = n 2$	1	4	9	16	25	36	49	64	81	100	$π$ ²/6^[2]	A000290
5	Pentagonal	$1 / 2 (3 n 2 - n)$	1	5	12	22	35	51	70	92	117	145	$3 ln 3 - π \sqrt 3 / 3$ ^[2]	A000326
6	Hexagonal	$1 / 2 (4 n 2 - 2 n) = 2 n 2 - n$	1	6	15	28	45	66	91	120	153	190	$2 ln 2$ ^[2]	A000384
7	Heptagonal	$1 / 2 (5 n 2 - 3 n)$	1	7	18	34	55	81	112	148	189	235	${\begin{matrix}{\tfrac {2}{3}}\ln 5\\+{\tfrac {{1}+{\sqrt {5}}}{3}}\ln {\tfrac {\sqrt {10-2{\sqrt {5}}}}{2}}\\+{\tfrac {{1}-{\sqrt {5}}}{3}}\ln {\tfrac {\sqrt {10+2{\sqrt {5}}}}{2}}\\+{\tfrac {\pi {\sqrt {25-10{\sqrt {5}}}}}{15}}\end{matrix}}$ ^[2]	A000566
8	Octagonal	$1 / 2 (6 n 2 - 4 n) = 3 n 2 - 2 n$	1	8	21	40	65	96	133	176	225	280	$3 / 4 ln 3 + π \sqrt 3 / 12$ ^[2]	A000567
9	Nonagonal	$1 / 2 (7 n 2 - 5 n)$	1	9	24	46	75	111	154	204	261	325		A001106
10	Decagonal	$1 / 2 (8 n 2 - 6 n) = 4 n 2 - 3 n$	1	10	27	52	85	126	175	232	297	370	$ln 2 + π / 6$	A001107
11	Hendecagonal	$1 / 2 (9 n 2 - 7 n)$	1	11	30	58	95	141	196	260	333	415		A051682
12	Dodecagonal	$1 / 2 (10 n 2 - 8 n)$	1	12	33	64	105	156	217	288	369	460		A051624
13	Tridecagonal	$1 / 2 (11 n 2 - 9 n)$	1	13	36	70	115	171	238	316	405	505		A051865
14	Tetradecagonal	$1 / 2 (12 n 2 - 10 n)$	1	14	39	76	125	186	259	344	441	550	$2 / 5 ln 2 + 3 / 10 ln 3 + π \sqrt 3 / 10$	A051866
15	Pentadecagonal	$1 / 2 (13 n 2 - 11 n)$	1	15	42	82	135	201	280	372	477	595		A051867
16	Hexadecagonal	$1 / 2 (14 n 2 - 12 n)$	1	16	45	88	145	216	301	400	513	640		A051868
17	Heptadecagonal	$1 / 2 (15 n 2 - 13 n)$	1	17	48	94	155	231	322	428	549	685		A051869
18	Octadecagonal	$1 / 2 (16 n 2 - 14 n)$	1	18	51	100	165	246	343	456	585	730	$4 / 7 ln 2 - \sqrt 2 / 14 ln (3 - 2 \sqrt 2)$ $+ π (1 + \sqrt 2) / 14$	A051870
19	Enneadecagonal	$1 / 2 (17 n 2 - 15 n)$	1	19	54	106	175	261	364	484	621	775		A051871
20	Icosagonal	$1 / 2 (18 n 2 - 16 n)$	1	20	57	112	185	276	385	512	657	820		A051872
21	Icosihenagonal	$1 / 2 (19 n 2 - 17 n)$	1	21	60	118	195	291	406	540	693	865		A051873
22	Icosidigonal	$1 / 2 (20 n 2 - 18 n)$	1	22	63	124	205	306	427	568	729	910		A051874
23	Icositrigonal	$1 / 2 (21 n 2 - 19 n)$	1	23	66	130	215	321	448	596	765	955		A051875
24	Icositetragonal	$1 / 2 (22 n 2 - 20 n)$	1	24	69	136	225	336	469	624	801	1000		A051876
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
10000	Myriagonal	$1 / 2 (9998 n 2 - 9996 n)$	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):

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 $s$ -gonal $t$ -gonal numbers for small values of $s$ and $t$ .

$s$	$t$	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

{{{annotations}}}

Proof without words that all hexagonal numbers are odd-sided triangular numbers

In some cases, such as $s = 10$ and $t = 4$ , 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.^[4]

The number 1225 is hecatonicositetragonal ( $s = 124$ ), hexacontagonal ( $s = 60$ ), icosienneagonal ( $s = 29$ ), hexagonal, square, and triangular.

Notes

1 2 Conway, John H.; Guy, Richard (2012-12-06). The Book of Numbers. Springer Science & Business Media. pp. 38–41. ISBN 978-1-4612-4072-3.
1 2 3 4 5 6 7 8 "Sums of Reciprocals of Polygonal Numbers and a Theorem of Gauss" (PDF). Archived from the original (PDF) on 2011-06-15. Retrieved 2010-06-13.
↑ "Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers" (PDF). Archived from the original (PDF) on 2013-05-29. Retrieved 2010-05-13.
↑ Weisstein, Eric W. "Pentagonal Square Triangular Number". MathWorld .

Related Research Articles

A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The $n$ th triangular number is the number of dots in the triangular arrangement with $n$ dots on each side, and is equal to the sum of the $n$ natural numbers from 1 to $n$ . The sequence of triangular numbers, starting with the 0th triangular number, is

In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals $32$ and can be written as $3 \times 3$ .

The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes and different dimensions. The term can mean

In Euclidean geometry, a regular polygon is a polygon that is direct equiangular and equilateral. Regular polygons may be either convex, star or skew. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon, if the edge length is fixed.

A hexagonal number is a figurate number. The nth hexagonal number h_n is the number of distinct dots in a pattern of dots consisting of the outlines of regular hexagons with sides up to n dots, when the hexagons are overlaid so that they share one vertex.

In mathematics and combinatorics, a centered hexagonal number, or hex number, is a centered figurate number that represents a hexagon with a dot in the center and all other dots surrounding the center dot in a hexagonal lattice. The following figures illustrate this arrangement for the first four centered hexagonal numbers:

A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The nth pentagonal number p_n is the number of distinct dots in a pattern of dots consisting of the outlines of regular pentagons with sides up to n dots, when the pentagons are overlaid so that they share one vertex. For instance, the third one is formed from outlines comprising 1, 5 and 10 dots, but the 1, and 3 of the 5, coincide with 3 of the 10 – leaving 12 distinct dots, 10 in the form of a pentagon, and 2 inside.

A pyramidal number is the number of points in a pyramid with a polygonal base and triangular sides. The term often refers to square pyramidal numbers, which have a square base with four sides, but it can also refer to a pyramid with any number of sides. The numbers of points in the base and in layers parallel to the base are given by polygonal numbers of the given number of sides, while the numbers of points in each triangular side is given by a triangular number. It is possible to extend the pyramidal numbers to higher dimensions.

<span class="mw-page-title-main">Tetrahedral number</span> Polyhedral number representing a tetrahedron

A tetrahedral number, or triangular pyramidal number, is a figurate number that represents a pyramid with a triangular base and three sides, called a tetrahedron. The $n$ th tetrahedral number, $Te n$ , is the sum of the first $n$ triangular numbers, that is,

An octagonal number is a figurate number that gives the number of points in a certain octagonal arrangement. The octagonal number for n is given by the formula 3n² − 2n, with n > 0. The first few octagonal numbers are

In mathematics, a pyramid number, or square pyramidal number, is a natural number that counts the stacked spheres in a pyramid with a square base. The study of these numbers goes back to Archimedes and Fibonacci. They are part of a broader topic of figurate numbers representing the numbers of points forming regular patterns within different shapes.

A pronic number is a number that is the product of two consecutive integers, that is, a number of the form $. The study of these numbers dates back to Aristotle. They are also called oblong numbers, heteromecic numbers, or rectangular numbers; however, the term "rectangular number" has also been applied to the composite numbers.$

In elementary number theory, a centered square number is a centered figurate number that gives the number of dots in a square with a dot in the center and all other dots surrounding the center dot in successive square layers. That is, each centered square number equals the number of dots within a given city block distance of the center dot on a regular square lattice. While centered square numbers, like figurate numbers in general, have few if any direct practical applications, they are sometimes studied in recreational mathematics for their elegant geometric and arithmetic properties.

The centered polygonal numbers are a class of series of figurate numbers, each formed by a central dot, surrounded by polygonal layers of dots with a constant number of sides. Each side of a polygonal layer contains one more dot than each side in the previous layer; so starting from the second polygonal layer, each layer of a centered k-gonal number contains k more dots than the previous layer.

A nonagonal number, or an enneagonal number, is a figurate number that extends the concept of triangular and square numbers to the nonagon. However, unlike the triangular and square numbers, the patterns involved in the construction of nonagonal numbers are not rotationally symmetrical. Specifically, the nth nonagonal number counts the dots in a pattern of n nested nonagons, all sharing a common corner, where the ith nonagon in the pattern has sides made of i dots spaced one unit apart from each other. The nonagonal number for n is given by the formula:

In number theory, a pentatope number is a number in the fifth cell of any row of Pascal's triangle starting with the 5-term row 1 4 6 4 1, either from left to right or from right to left. It is named because it represents the number of 3-dimensional unit spheres which can be packed into a pentatope of increasing side lengths.

A centered decagonal number is a centered figurate number that represents a decagon with a dot in the center and all other dots surrounding the center dot in successive decagonal layers. The centered decagonal number for n is given by the formula

In geometry of 4 dimensions or higher, a double prism or duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an $n$ -polytope and an $m$ -polytope is an $(n + m)$ -polytope, where $n$ and $m$ are dimensions of 2 (polygon) or higher.

In the mathematics of figurate numbers, the cannonball problem asks which numbers are both square and square pyramidal. The problem can be stated as: given a square arrangement of cannonballs, for what size squares can these cannonballs also be arranged into a square pyramid. Equivalently, which squares can be represented as the sum of consecutive squares, starting from 1.

A centered octahedral number or Haüy octahedral number is a figurate number that counts the points of a three-dimensional integer lattice that lie inside an octahedron centered at the origin. The same numbers are special cases of the Delannoy numbers, which count certain two-dimensional lattice paths. The Haüy octahedral numbers are named after René Just Haüy.

References

The Penguin Dictionary of Curious and Interesting Numbers , David Wells (Penguin Books, 1997) [ ISBN 0-14-026149-4].
Polygonal numbers at PlanetMath
Weisstein, Eric W. "Polygonal Numbers". MathWorld .
F. Tapson (1999). The Oxford Mathematics Study Dictionary (2nd ed.). Oxford University Press. pp. 88–89. ISBN 0-19-914-567-9.

External links

"Polygonal number", Encyclopedia of Mathematics , EMS Press, 2001 [1994]
Polygonal Numbers: Every s-polygonal number between 1 and 1000 clickable for 2<=s<=337
Polygonal Numbers on the Ulam Spiral grid on YouTube
Polygonal Number Counting Function: http://www.mathisfunforum.com/viewtopic.php?id=17853

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[:0-1] 1 2 Conway, John H.; Guy, Richard (2012-12-06). The Book of Numbers. Springer Science & Business Media. pp. 38–41. ISBN 978-1-4612-4072-3.

[siam_07-003s-2] 1 2 3 4 5 6 7 8 "Sums of Reciprocals of Polygonal Numbers and a Theorem of Gauss" (PDF). Archived from the original (PDF) on 2011-06-15. Retrieved 2010-06-13.

[3] "Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers" (PDF). Archived from the original (PDF) on 2013-05-29. Retrieved 2010-05-13.

[4] Weisstein, Eric W. "Pentagonal Square Triangular Number". MathWorld .

[1]

[2]

[3]

[4]