Centered Polygonal Number

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Instructions

The centered polygonal numbers are a family of sequences of 2-dimensional figurate numbers, each formed by a central dot, sorrounded by polygonal layers with a constant number of sides. Each side of a polygonal layer contains one dot more than a side in the previous layer.

| - Centered triangular numbers | Centered square numbers | Centered pentagonal numbers | Centered hexagonal numbers

In the following table are listed the first 12 terms of the sequences of centered k-polygonal numbers, with k from 3 to 22:

k	Name	0	1	2	3	4	5	6	7	8	9	10	11
3	Triangular	1	4	10	19	31	46	64	85	109	136	166	199
4	Square	1	5	13	25	41	61	85	113	145	181	221	265
5	Pentagonal	1	6	16	31	51	76	106	141	181	226	276	331
6	Hexagonal	1	7	19	37	61	91	127	169	217	271	331	397
7	Heptagonal	1	8	22	43	71	106	148	197	253	316	386	463
8	Octagonal	1	9	25	49	81	121	169	225	289	361	441	529
9	Nonagonal	1	10	28	55	91	136	190	253	325	406	496	595
10	Decagonal	1	11	31	61	101	151	211	281	361	451	551	661
11	Hendecagonal	1	12	34	67	111	166	232	309	397	496	606	727
12	Dodecagonal	1	13	37	73	121	181	253	337	433	541	661	793
13	Tridecagonal	1	14	40	79	131	196	274	365	469	586	716	859
14	Tetradecagonal	1	15	43	85	141	211	295	393	505	631	771	925
15	Pentadecagonal	1	16	46	91	151	226	316	421	541	676	826	991
16	Hexadecagonal	1	17	49	97	161	241	337	449	577	721	881	1057
17	Heptadecagonal	1	18	52	103	171	256	358	477	613	766	936	1123
18	Octadecagonal	1	19	55	109	181	271	379	505	649	811	991	1189
19	Enneadecagonal	1	20	58	115	191	286	400	533	685	856	1046	1255
20	Icosagonal	1	21	61	121	201	301	421	561	721	901	1101	1321
21	Icosihenagonal	1	22	64	127	211	316	442	589	757	946	1156	1387
22	Icosidigonal	1	23	67	133	221	331	463	617	793	991	1211	1453

As you can see:

6 is the 1st pentagonal number;
16 is the 2nd pentagonal number and the 1st pentadecagonal number;
19 is the 3rd triangular number, the 2nd hexagonal number and the 1st octadecagonal number.

Write a function that takes an integer n as argument and returns its classification as polygonal number:

return "0th of all" if n is 1, since it is the 0th term of every centered polygonal number sequence;
return a list whose generic element is a string formatted as "{i}th {k}-gonal number" if n is the i-th k-gonal number, in k-ascending order;
return false if n is not a k-gonal number for any k > 2.

Examples

isPolygonal(3) ➞ false

isPolygonal(4) ➞ ["1st 3-gonal number"]

isPolygonal(16) ➞ ["2nd 5-gonal number", "1st 15-gonal number"]

Notes

N/A

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