Polygonal number

In mathematics, a polygonal number is a number that counts dots arranged in the shape of a regular polygon^[1]^: 2-3. 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^[1]^: 1.

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

An s-gonal number can be decomposed into s−2 triangular numbers and a natural number.

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:^[2]

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

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

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:^[2]

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.^[3]

$s$	Name	Formula	$n$										Sum of reciprocals^[3]^[4]	OEIS number
$s$	Name	Formula	1	2	3	4	5	6	7	8	9	10	Sum of reciprocals^[3]^[4]	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^[3]	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^[3]	A000290
5	Pentagonal	$⁠ 1 / 2 ⁠ (3 n 2 - n)$	1	5	12	22	35	51	70	92	117	145	$3 ln 3 - ⁠ π \sqrt 3 / 3 ⁠$ ^[3]	A000326
6	Hexagonal	$⁠ 1 / 2 ⁠ (4 n 2 - 2 n) = 2 n 2 - n$	1	6	15	28	45	66	91	120	153	190	$2 ln 2$ ^[3]	A000384
7	Heptagonal	$⁠ 1 / 2 ⁠ (5 n 2 - 3 n)$	1	7	18	34	55	81	112	148	189	235	${\begin{matrix}{\tfrac {2}{3}}\ln 5\\+{\tfrac {{1}+{\sqrt {5}}}{3}}\ln {\tfrac {\sqrt {10-2{\sqrt {5}}}}{2}}\\+{\tfrac {{1}-{\sqrt {5}}}{3}}\ln {\tfrac {\sqrt {10+2{\sqrt {5}}}}{2}}\\+{\tfrac {\pi {\sqrt {25-10{\sqrt {5}}}}}{15}}\end{matrix}}$ ^[3]	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	$⁠ 3 / 4 ⁠ ln 3 + ⁠ π \sqrt 3 / 12 ⁠$ ^[3]	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

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.^[5]

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

Notes

^ ^a ^b Tattersall, James J. (2005). Elementary Number Theory in Nine Chapters (2nd ed.). New York: Cambridge University Press. ISBN 978-0-511-75634-4.
^ ^a ^b 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.
^ ^a ^b ^c ^d ^e ^f ^g ^h "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.

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

[tattersall2005-1] Tattersall, James J. (2005). Elementary Number Theory in Nine Chapters (2nd ed.). New York: Cambridge University Press. ISBN 978-0-511-75634-4.

[:0-2] 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-3] ^ ^a ^b ^c ^d ^e ^f ^g ^h "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.

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

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

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