Polygonal number

In mathematics, a polygonal number is a number represented as dots or pebbles arranged in the shape of a regular polygon. The dots are thought of as alphas (units). 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{n^2(s-2)-n(s-4)}{2}

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:

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

Thus:

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}\, .

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)}.

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}}=n(n-1)/2

it follows that:

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

This shows that the $n^{th}$ hexagonal number, $P(6,n)$ is the $(2n-1)^{{st}}$ triangular number, $T_{{2n-1}}$ . 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.^[1]

s	Name	Formula	n = 1	n = 2	n = 3	n = 4	n = 5	n = 6	n = 7	n = 8	n = 9	n = 10	Sum of Reciprocals^[1]^[2]	OEIS number
3	Triangular	½(n²+n)	1	3	6	10	15	21	28	36	45	55	${2}$ ^[1]	A000217
4	Square	n² = ½(2n² - 0n)	1	4	9	16	25	36	49	64	81	100	${\pi^2\over6}$ ^[1]	A000290
5	Pentagonal	½(3n² - n)	1	5	12	22	35	51	70	92	117	145	${ 3\ln\left(3\right)}-{\pi\sqrt{3}\over3 }$ ^[1]	A000326
6	Hexagonal	½(4n² - 2n)	1	6	15	28	45	66	91	120	153	190	${ 2\ln\left(2\right) }$ ^[1]	A000384
7	Heptagonal	½(5n² - 3n)	1	7	18	34	55	81	112	148	189	235	${\begin{matrix}{\frac {2}{3}}\ln(5)\\+{\frac {{1}+{\sqrt {5}}}{3}}\ln \left({\frac {1}{2}}{\sqrt {10-2{\sqrt {5}}}}\right)\\+{\frac {{1}-{\sqrt {5}}}{3}}\ln \left({\frac {1}{2}}{\sqrt {10+2{\sqrt {5}}}}\right)\\+{\frac {1}{15}}{\pi }{{\sqrt {25-10{\sqrt {5}}}}}\end{matrix}}$ ^[1]	A000566
8	Octagonal	½(6n² - 4n)	1	8	21	40	65	96	133	176	225	280	${ {3\ln\left(3\right)\over4} + {\pi\sqrt{3}\over12} }$ ^[1]	A000567
9	Nonagonal (enneagonal)	½(7n² - 5n)	1	9	24	46	75	111	154	204	261	325		A001106
10	Decagonal	½(8n² - 6n)	1	10	27	52	85	126	175	232	297	370	${ {\ln\left(2\right)} + {\pi\over6} }$	A001107
11	Hendecagonal	½(9n² - 7n)	1	11	30	58	95	141	196	260	333	415		A051682
12	Dodecagonal	½(10n² - 8n)	1	12	33	64	105	156	217	288	369	460		A051624
13	Tridecagonal	½(11n² - 9n)	1	13	36	70	115	171	238	316	405	505		A051865
14	Tetradecagonal	½(12n² - 10n)	1	14	39	76	125	186	259	344	441	550	${ {2\ln\left(2\right)\over5} + {3\ln\left(3\right)\over 10} + {\pi\sqrt{3}\over10} }$	A051866
15	Pentadecagonal	½(13n² - 11n)	1	15	42	82	135	201	280	372	477	595		A051867
16	Hexadecagonal	½(14n² - 12n)	1	16	45	88	145	216	301	400	513	640		A051868
17	Heptadecagonal	½(15n² - 13n)	1	17	48	94	155	231	322	428	549	685		A051869
18	Octadecagonal	½(16n² - 14n)	1	18	51	100	165	246	343	456	585	730	${\begin{matrix}{\frac {4}{7}}\ln {\left(2\right)}\\-{\frac {{\sqrt {2}}}{14}}\ln \left(3-2{\sqrt {2}}\right)\\+\pi {\frac {\left(1+{\sqrt {2}}\right)}{14}}\end{matrix}}$	A051870
19	Enneadecagonal	½(17n² - 15n)	1	19	54	106	175	261	364	484	621	775		A051871
20	Icosagonal	½(18n² - 16n)	1	20	57	112	185	276	385	512	657	820		A051872
21	Icosihenagonal	½(19n² - 17n)	1	21	60	118	195	291	406	540	693	865		A051873
22	Icosidigonal	½(20n² - 18n)	1	22	63	124	205	306	427	568	729	910		A051874
23	Icositrigonal	½(21n² - 19n)	1	23	66	130	215	321	448	596	765	955		A051875
24	Icositetragonal	½(22n² - 20n)	1	24	69	136	225	336	469	624	801	1000		A051876
25	Icosipentagonal	½(23n² - 21n)	1	25	72	142	235	351	490	652	837	1045		A255184
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
10000	Myriagonal	½(9998n² - 9996n)	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	Odd triangular square numbers.	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, 4835857349623374369, 571178264921393749929, 4342594514813297471521, 512917445842648529510881, …	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
10	3	1, 10, 1540, 1777555, 13773376, 2051297326, 15894464365, 2367195337045, 18342198104230, ...
10	4	1 and no other.
11	4	1, 196, 29241, 1755625, 261468900, 38941102225, 2337990844401, 348201795147556, 51858411008887561, 3113535139359330841, 463705205422871375236, 69060571958250748760481, 4146338334574433921200225, ...
12	3	1, 105, 561, 41041, 3000025, 219294153, 1167483681,...
12	4	1, 64, 3025, 142129, 6677056, 313679521, 14736260449, 692290561600, 32522920134769, 1527884955772561, 71778070001175616, 3372041405099481409, 158414167969674450625, ...
12	5	1, 12, 324105, 2884960, 75296819577, ...
13	3	1, 36, 171, 4186, 237016, 5785101, 27169506, 663153571, 37543891231, ...
13	4	1, 36, 35721, 34999056, 896703025, 34291262041, 878568782400, 860801272542225, ...
14	3	1, 1275, 58311, 48005301, 2194498125, ...
14	4	1, 441, 14161, 4239481, 135978921, 40707501121, 1305669590281, 390873421529361, 12537039269904241, 3753166552817428201, ...
15	3	1, 15, 595, 221445, 863955, 321197185, 12232747905, 119413111951, 4547830928851, 44394820201725, ...
15	4	1, 3025, 5997601, 165148201, 327372787225, 648950099301001, 17869299978562201, ...
16	3	1, 45, 1485, 41041, 1334161, 36855405, 1198075725, ...
16	4	1, 16, 400, 4225, 101761, 1073296, 25847056, 272613121, 6565050625, ...
18	4	1, 100, 1936, 116281, 2235025, 134189056, 2579217796, 154854055225, 2976415102441, ...
21	3	1, 21, 406, 30381, 86736, 6474601, 119761026, 1379779246, ...
21	4	1, 2601, 3705625, 54745201, ...
22	3	1, 11935, 203203, 1237556125, 21068604901, ...
22	4	1, 729, 284089, 3900625, 15175959521, 590725976569, 8110813506601, 3156387347610225, 1228333148092290241, 16865317394711073289, 6563271907899976822281, 2554149271482890096235025, 35069100108493095964960369, ...
23	3	1, 66, 558096, 18644671, 294116131, 9825706836, ...
24	3	1, 136, 2016, 51681, 134421, 3444000, 50637016, 3374352325, ...
24	4	1, 225, 25921, 312481, 35820225, 4106118241, ...
25	3	1, 351, 4851, 810901, 11169901, 1866696651, ...
25	4	1, 25, 8100, 104976, 32478601, 420865225, 4786564225, 62025404401, ...
28	3	1, 28, 4465, 10585, 1606528, 20985481, 243796321, 3184618528, ...
28	4	1, 81, 3136, 30625, 1166400, 44422225, 433638976, 16515077121, ...
29	3	1, 1225, 2850, 1519896, 16799706, 8954498400, ...
29	4	1, 1225, 680625, 377719225, 209618381281, ...
30	3	1, 595, 3649051, 926844985, 9775585225, ...
30	4	1, 203401, 1819801, 164024190001, 1467492382801, 132269434866199801, 1183388792474889001, 106662336814809228952801, 954287089027867949018401, 86012721732003522411131649001, 769539017165067381031862931001, ...
32	4	1, 1089, 9025, 4190209, 34680321, 16098788161, 133241790529, 61851539930625, 511914924538369, 237633600314679361, 1966777006834629441, 912988230557458180609, 7556356748343721780225, ...
34	3	1, 325, 2926, 377146, 3378700, 435228256, ...
36	3	1, 36, 105, 1540, 12880, 185745, 523776, 7552441, 63095761, 909788496, 2565464265, ...
36	4	1, 36, 4096, 443556, 6630625, 48011041, 717704100, 77684838400, 8408666052900, 125698875825601, 910160415388225, ...
40	4	1, 576, 123201, 26255376, 5595189601, ...
41	3	1, 120, 57970, 2478651, 17841351, 762822270, ...
44	4	1, 256, 1521, 136161, 802816, 71757841, 423083761, 37816247296, 222964340481, ...
45	3	1, 45, 435, 2850, 44551, 41810940, 79714251, ...
49	3	1, 4950, 29646, 45634681, 273183625, ...
49	4	1, 49, 144, 12996, 613089, 54331641, 2562586884, 227094183936, 644826666121, ...
50	3	1, 20910, 122760, 787032975, 4620055875, ...
50	4	1, 5776, 30276, 55487601, 290736601, 532791965476, 2791652838976, 5115868397039401, 26805450269137401, 49122567815580389376, 257385930692604511876, 471674891049334501775401, 2471419679704938253922401, ...

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 appear in print.^[3]

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

The only polygonal set that is contained in another polygonal set is the set of hexagonal numbers, which is contained in the set of triangular numbers.

Notes

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

Hazewinkel, Michiel, ed. (2001), "Polygonal number", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
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

Classes of natural numbers

Powers and
related numbers

Of the form
a × 2^b ± 1

Other polynomial
numbers

Recursively defined
numbers

Possessing a
specific set
of other numbers

Expressible via
specific sums

Generated
via a sieve

Lucky

Code related

Meertens

Figurate numbers

2-dimensional

centered	Centered triangular Centered square Centered pentagonal Centered hexagonal Centered heptagonal Centered octagonal Centered nonagonal Centered decagonal Star

non-centered	Triangular Square Square triangular Pentagonal Hexagonal Heptagonal Octagonal Nonagonal Decagonal Dodecagonal

3-dimensional

centered	Centered tetrahedral Centered cube Centered octahedral Centered dodecahedral Centered icosahedral

non-centered	Tetrahedral Octahedral Dodecahedral Icosahedral Stella octangula

pyramidal	Square pyramidal Pentagonal pyramidal Hexagonal pyramidal Heptagonal pyramidal

4-dimensional

centered	Centered pentachoric Squared triangular

non-centered	Pentatope

Pseudoprimes

Combinatorial
numbers

Arithmetic functions

By properties of σ(n)	Abundant Almost perfect Arithmetic Colossally abundant Descartes Hemiperfect Highly abundant Highly composite Hyperperfect Multiply perfect Perfect Practical number Primitive abundant Quasiperfect Refactorable Sublime Superabundant Superior highly composite Superperfect

By properties of Ω(n)	Almost prime Semiprime

By properties of φ(n)	Highly cototient Highly totient Noncototient Nontotient Perfect totient Sparsely totient

By properties of s(n)	Amicable Betrothed Deficient Semiperfect

Dividing a quotient

Other prime factor
or divisor related
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Recreational
mathematics

Base-dependent
numbers

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