Polygonal number

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In mathematics, a polygonal number is a number that can be arranged as a regular polygon. Ancient mathematicians discovered that numbers could be arranged in certain ways when they were represented by pebbles or seeds; such numbers, which can be made from figures, are generally called figurate numbers.

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 ${(s-2)n^2-(s-4)n}\over 2$ .

Name	Formula	n=1	2	3	4	5	6	7	8	9	10	11	12	13
Triangular	½(1n² + 1n)	1	3	6	10	15	21	28	36	45	55	66	78	91
Square	½(2n² - 0n)	1	4	9	16	25	36	49	64	81	100	121	144	169
Pentagonal	½(3n² - 1n)	1	5	12	22	35	51	70	92	117	145	176	210	247
Hexagonal	½(4n² - 2n)	1	6	15	28	45	66	91	120	153	190	231	276	325
Heptagonal	½(5n² - 3n)	1	7	18	34	55	81	112	148	189	235	286	342	403
Octagonal	½(6n² - 4n)	1	8	21	40	65	96	133	176	225	280	341	408	481
Nonagonal	½(7n² - 5n)	1	9	24	46	75	111	154	204	261	325	396	474	559
Decagonal	½(8n² - 6n)	1	10	27	52	85	126	175	232	297	370	451	540	637
Hendecagonal	½(9n² - 7n)	1	11	30	58	95	141	196	260	333	415	506	606	715
Dodecagonal	½(10n² - 8n)	1	12	33	64	105	156	217	288	369	460	561	672	793
Tridecagonal	½(11n² - 9n)	1	13	36	70	115	171	238	316	405	505	616	738	871
Tetradecagonal	½(12n² - 10n)	1	14	39	76	125	186	259	344	441	550	671	804	949
Pentadecagonal	½(13n² - 11n)	1	15	42	82	135	201	280	372	477	595	726	870	1027
Hexadecagonal	½(14n² - 12n)	1	16	45	88	145	216	301	400	513	640	781	936	1105
Heptadecagonal	½(15n² - 13n)	1	17	48	94	155	231	322	428	549	685	836	1002	1183
Octadecagonal	½(16n² - 14n)	1	18	51	100	165	246	343	456	585	730	891	1068	1261
Nonadecagonal	½(17n² - 15n)	1	19	54	106	175	261	364	484	621	775	946	1134	1339
Icosagonal	½(18n² - 16n)	1	20	57	112	185	276	385	512	657	820	1001	1200	1417
Icosihenagonal	½(19n² - 17n)	1	21	60	118	195	291	406	540	693	865	1056	1266	1495
Icosidigonal	½(20n² - 18n)	1	22	63	124	205	306	427	568	729	910	1111	1332	1573
Icositrigonal	½(21n² - 19n)	1	23	66	130	215	321	448	596	765	955	1166	1398	1651
Icositetragonal	½(22n² - 20n)	1	24	69	136	225	336	469	624	801	1000	1221	1464	1729
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

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