Polygonal number: Information from Answers.com

In mathematics, a polygonal number is a number represented as dots or pebbles arrayed in the shape of a polygon. The dots were thought of as alphas (units). These are one type of figurate numbers.

1 Definition and examples
2 Combinations
3 Notes
4 References
5 External links

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 triangular square 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

1		3		6		10

Square numbers

1		4		9		16

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 regular lattice like above. For example, the first few hexagonal numbers are:

1		6		15		28

If s is the number of sides in a polygon, the formula for the n^th s-gonal number is ${(\frac{s}{2}-1)n^2-(\frac{s}{2}-2)n}$ .

Name	Formula	n=1	2	3	4	5	6	7	8	9	10	11	12	13	Sum of Reciprocals^[1]	OEIS number
Triangular	½(1n² + 1n)	1	3	6	10	15	21	28	36	45	55	66	78	91	2	A000217
Square	½(2n² - 0n)	1	4	9	16	25	36	49	64	81	100	121	144	169	${\pi^2 \over 6}$	A000290
Pentagonal	½(3n² - 1n)	1	5	12	22	35	51	70	92	117	145	176	210	247	$3\ln\left(3\right)-{\pi\sqrt{3}\over3}$	A000326
Hexagonal	½(4n² - 2n)	1	6	15	28	45	66	91	120	153	190	231	276	325	$2\ln\left(2\right)$	A000384
Heptagonal	½(5n² - 3n)	1	7	18	34	55	81	112	148	189	235	286	342	403		A000566
Octagonal	½(6n² - 4n)	1	8	21	40	65	96	133	176	225	280	341	408	481	${3\ln\left(3\right)\over 4}+{\sqrt{3}\pi\over 12}$	A000567
Nonagonal	½(7n² - 5n)	1	9	24	46	75	111	154	204	261	325	396	474	559		A001106
Decagonal	½(8n² - 6n)	1	10	27	52	85	126	175	232	297	370	451	540	637	$2\ln\left(2\right)+{\pi \over 6}$	A001107
Hendecagonal	½(9n² - 7n)	1	11	30	58	95	141	196	260	333	415	506	606	715		A051682
Dodecagonal	½(10n² - 8n)	1	12	33	64	105	156	217	288	369	460	561	672	793		A051624
Tridecagonal	½(11n² - 9n)	1	13	36	70	115	171	238	316	405	505	616	738	871		A051865
Tetradecagonal	½(12n² - 10n)	1	14	39	76	125	186	259	344	441	550	671	804	949	${2\ln\left(2\right)\over 5}+{3\ln\left(3\right)\over 10}+{\sqrt{3}\pi\over 10}$	A051866
Pentadecagonal	½(13n² - 11n)	1	15	42	82	135	201	280	372	477	595	726	870	1027		A051867
Hexadecagonal	½(14n² - 12n)	1	16	45	88	145	216	301	400	513	640	781	936	1105		A051868
Heptadecagonal	½(15n² - 13n)	1	17	48	94	155	231	322	428	549	685	836	1002	1183		A051869
Octadecagonal	½(16n² - 14n)	1	18	51	100	165	246	343	456	585	730	891	1068	1261		A051870
Nonadecagonal	½(17n² - 15n)	1	19	54	106	175	261	364	484	621	775	946	1134	1339		A051871
Icosagonal	½(18n² - 16n)	1	20	57	112	185	276	385	512	657	820	1001	1200	1417		A051872
Icosihenagonal	½(19n² - 17n)	1	21	60	118	195	291	406	540	693	865	1056	1266	1495		A051873
Icosidigonal	½(20n² - 18n)	1	22	63	124	205	306	427	568	729	910	1111	1332	1573		A051874
Icositrigonal	½(21n² - 19n)	1	23	66	130	215	321	448	596	765	955	1166	1398	1651		A051875
Icositetragonal	½(22n² - 20n)	1	24	69	136	225	336	469	624	801	1000	1221	1464	1729		A051876
Icosipentagonal	½(23n² - 21n)	1	25	72	142	235	351	490	652	837	1045	1276	1530	1807
Icosihexagonal	½(24n² - 22n)	1	26	75	148	245	366	511	680	873	1090	1331	1596	1885
Icosiheptagonal	½(25n² - 23n)	1	27	78	154	255	381	532	708	909	1135	1386	1662	1963
Icosioctagonal	½(26n² - 24n)	1	28	81	160	265	396	553	736	945	1180	1441	1728	2041
Icosinonagonal	½(27n² - 25n)	1	29	84	166	275	411	574	764	981	1225	1496	1794	2119
Triacontagonal	½(28n² - 26n)	1	30	87	172	285	426	595	792	1017	1270	1551	1860	2197

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").

For a given s-gonal number x, one can find n by

$n = \frac{\sqrt{(8s-16)x+(s-4)^2}+s-4}{2s-4}.$

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	Ref.
4	3	1, 36, 1225, 41616, …	A001110	^[2]
5	3	1, 210, 40755, 7906276, …	A014979	^[3]
5	4	1, 9801, 94109401, …	A036353	^[4]
6	3	All hexagonal numbers are also triangular.
6	4	1, 1225, 1413721, 1631432881, …	A046177	^[5]
6	5	1, 40755, 1533776805, …	A046180	^[6]
7	3	1, 55, 121771, 5720653, …	A046194	^[7]
7	4	1, 81, 5929, 2307361, …	A036354	^[8]
7	5	1, 4347, 16701685, 64167869935, …	A048900	^[9]
7	6	1, 121771, 12625478965, …	A048903	^[10]
8	3	1, 21, 11781, 203841, …	A046183	^[11]
8	4	1, 225, 43681, 8473921, …	A036428	^[12]
8	5	1, 176, 1575425, 234631320, …	A046189	^[13]
8	6	1, 11781, 113123361, …	A046192	^[14]
8	7	1, 297045, 69010153345, …	A048906	^[15]
9	3	1, 325, 82621, 20985481, …	A048909	^[16]
9	4	1, 9, 1089, 8281, 978121, …	A036411	^[17]
9	5	1, 651, 180868051, …	A048915	^[18]
9	6	1, 325, 5330229625, …	A048918	^[19]
9	7	1, 26884, 542041975, …	A048921	^[20]
9	8	1, 631125, 286703855361, …	A048924	^[21]

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 such number has yet to appear in print.^[22] All hexagonal square numbers are also hexagonal square triangular numbers, and 1225 is actually a hecticositetragonal, hexacontagonal, icosinonagonal, hexagonal, square, triangular number.

Notes

^ Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers
^ Weisstein, Eric W., "Square Triangular Number" from MathWorld.
^ Weisstein, Eric W., "Pentagonal Triangular Number" from MathWorld.
^ Weisstein, Eric W., "Pentagonal Square Number" from MathWorld.
^ Weisstein, Eric W., "Hexagonal Square Number" from MathWorld.
^ Weisstein, Eric W., "Hexagonal Pentagonal Number" from MathWorld.
^ Weisstein, Eric W., "Heptagonal Triangular Number" from MathWorld.
^ Weisstein, Eric W., "Heptagonal Square Number" from MathWorld.
^ Weisstein, Eric W., "Heptagonal Pentagonal Number" from MathWorld.
^ Weisstein, Eric W., "Heptagonal Hexagonal Number" from MathWorld.
^ Weisstein, Eric W., "Octagonal Triangular Number" from MathWorld.
^ Weisstein, Eric W., "Octagonal Square Number" from MathWorld.
^ Weisstein, Eric W., "Octagonal Pentagonal Number" from MathWorld.
^ Weisstein, Eric W., "Octagonal Hexagonal Number" from MathWorld.
^ Weisstein, Eric W., "Octagonal Heptagonal Number" from MathWorld.
^ Weisstein, Eric W., "Nonagonal Triangular Number" from MathWorld.
^ Weisstein, Eric W., "Nonagonal Square Number" from MathWorld.
^ Weisstein, Eric W., "Nonagonal Pentagonal Number" from MathWorld.
^ Weisstein, Eric W., "Nonagonal Hexagonal Number" from MathWorld.
^ Weisstein, Eric W., "Nonagonal Heptagonal Number" from MathWorld.
^ Weisstein, Eric W., "Nonagonal Octagonal Number" from MathWorld.
^ Weisstein, Eric W., "Pentagonal Square Triangular Number" from 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" from MathWorld.

External links

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

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Polygonal number

Contents

Definition and examples

Combinations

Notes

References

External links