Figurate numbers

Since at least the time of the ancient Greeks, man has studied sequences of numbers which correspond to the geometric arrangement of like items (e.g. dots, pebbles, atoms, ...) which begets the figurate numbers. For the Greeks of Antiquity, these numbers related either to 2-dimensional figures such as polygons, or to 3-dimensional figures such as pyramids. Figurate numbers, which are usually associated with the 2 or 3-dimensional figures, may be generalized to higher dimensions where they are often called polytope numbers and also to lower dimensions (where they are called gnomonic numbers and centered gnomonic numbers for 1-dimensional figures).

This introductory page shows only the 2-dimensional regular convex figurate numbers (polygonal numbers and centered polygonal numbers) and their associated 3-dimensional pyramidal layerings (pyramidal numbers and centered pyramidal numbers, i.e. (centered polygons) pyramidal numbers), which are not globally centered (only the original polygons are). Although pyramids (with any polygonal base) are not regular polyhedrons (with the exception of triangular pyramids, which are tetrahedrons, one of the 5 Platonic solids), the Greeks had a particular fascination towards square pyramids. It turns out that a square dipyramid is a regular polyhedron, an octahedron, one of the 5 Platonic solids!

All figurate numbers are fully classified on the page: Classifications of figurate numbers.

Two-dimensional figurate numbers

Polygonal numbers

Triangular numbers	Square numbers	Pentagonal numbers	Hexagonal numbers

The polygonal numbers

P  (2) V(n)

model

V

-sided (the number of sides of a polygon being equal to its number of vertices

V

) regular convex polygons which grow by the addition of layers to

V  −  2

of the sides.

The general formula for regular convex polygonal numbers is:

P  (2) V(n)  =  (V − 2) n 2 − (V − 4) n 2  =  n 2 [(V − 2) n − (V − 4)],

where

V

is the number of vertices (thus sides) of the polygon. Prime numbers (shown in bold) occur only in

P  (2) V(2) = V

since

P  (2) V(n), n   ≥   3,

is obviously composite from the formula.

See Polygonal numbers (and Category:Polygonal numbers) for more information (e.g. more formulae, recurrence equations, generating functions, order of basis, differences, partial sums, partial sums of reciprocals, sum of reciprocals) about the regular polygonal numbers.

Polygonal numbers formulae and values

A-number	V -gonal numbers	Formula P  (2) V(n)	n = 0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
A000217  (n)	Trigonal numbers (Triangular numbers)	${\frac {n^{2}+n}{2}}\,$	0	1	3	6	10	15	21	28	36	45	55	66	78	91	105	120	136	153	171	190	210
A000290  (n)	Tetragonal numbers (Square numbers)	$n^{2}\,$	0	1	4	9	16	25	36	49	64	81	100	121	144	169	196	225	256	289	324	361	400
A000326  (n)	Pentagonal numbers	${\frac {3n^{2}-n}{2}}\,$	0	1	5	12	22	35	51	70	92	117	145	176	210	247	287	330	376	425	477	532	590
A000384  (n)	Hexagonal numbers	$2n^{2}-n\,$	0	1	6	15	28	45	66	91	120	153	190	231	276	325	378	435	496	561	630	703	780
A000566  (n)	Heptagonal numbers	${\frac {5n^{2}-3n}{2}}\,$	0	1	7	18	34	55	81	112	148	189	235	286	342	403	469	540	616	697	783	874	970
A000567  (n)	Octagonal numbers	$3n^{2}-2n\,$	0	1	8	21	40	65	96	133	176	225	280	341	408	481	560	645	736	833	936	1045	1160
A001106  (n)	Nonagonal numbers	${\frac {7n^{2}-5n}{2}}\,$	0	1	9	24	46	75	111	154	204	261	325	396	474	559	651	750	856	969	1089	1216	1350
A001107  (n)	Decagonal numbers	$4n^{2}-3n\,$	0	1	10	27	52	85	126	175	232	297	370	451	540	637	742	855	976	1105	1242	1387	1540
A051682  (n)	Hendecagonal numbers	${\frac {9n^{2}-7n}{2}}\,$	0	1	11	30	58	95	141	196	260	333	415	506	606	715	833	960	1096	1241	1395	1558	1730
A051624  (n)	Dodecagonal numbers	$5n^{2}-4n\,$	0	1	12	33	64	105	156	217	288	369	460	561	672	793	924	1065	1216	1377	1548	1729	1920
A051865  (n)	Tridecagonal numbers	${\frac {11n^{2}-9n}{2}}\,$	0	1	13	36	70	115	171	238	316	405	505	616	738	871	1015	1170	1336	1513	1701	1900	2110
A051866  (n)	Tetradecagonal numbers	$6n^{2}-5n\,$	0	1	14	39	76	125	186	259	344	441	550	671	804	949	1106	1275	1456	1649	1854	2071	2300
A051867  (n)	Pentadecagonal numbers	${\frac {13n^{2}-11n}{2}}\,$	0	1	15	42	82	135	201	280	372	477	595	726	870	1027	1197	1380	1576	1785	2007	2242	2490
A051868  (n)	Hexadecagonal numbers	$7n^{2}-6n\,$	0	1	16	45	88	145	216	301	400	513	640	781	936	1105	1288	1485	1696	1921	2160	2413	2680
A051869  (n)	Heptadecagonal numbers	${\frac {15n^{2}-13n}{2}}\,$	0	1	17	48	94	155	231	322	428	549	685	836	1002	1183	1379	1590	1816	2057	2313	2584	2870
A051870  (n)	Octadecagonal numbers	$8n^{2}-7n\,$	0	1	18	51	100	165	246	343	456	585	730	891	1068	1261	1470	1695	1936	2193	2466	2755	3060
A051871  (n)	Nonadecagonal numbers	${\frac {17n^{2}-15n}{2}}\,$	0	1	19	54	106	175	261	364	484	621	775	946	1134	1339	1561	1800	2056	2329	2619	2926	3250
A051872  (n)	Icosagonal numbers	$9n^{2}-8n\,$	0	1	20	57	112	185	276	385	512	657	820	1001	1200	1417	1652	1905	2176	2465	2772	3097	3440

Centered polygonal numbers

Centered triangular numbers	Centered square numbers	Centered pentagonal numbers	Centered hexagonal numbers (Hex numbers)

The centered polygonal numbers

c P  (2) V(n)

model

V

-sided regular convex polygons (the number of sides of a polygon being equal to its number of vertices

V

) centered on a single dot (for

n   ≥   0

), in which each successive layer surrounds the previous layer.

The general formula for centered polygonal numbers is:

c P  (2) V(n)  =  V P  (2) 3(n) + 1  =  V Tn + 1  =  V  (  n + 12  )  + 1  =  V  n  (n + 1) 2 + 1,

where

P  (2) 3(n) = Tn

is the

n

th triangular number.

Prime numbers (shown in bold) occur often among the centered polygonal numbers due to the 1 addend in the formula. There are primes among all the centered polygonal numbers from the centered triangular numbers to the centered icosagonal numbers, with the notable exceptions of the centered octagonal numbers (which can never be prime since they are odd squares) and the centered nonagonal numbers (which can never be prime since they are a subsequence of the triangular numbers).

See Centered polygonal numbers (and Category:Centered polygonal numbers) for more information (e.g. more formulae, recurrence equations, generating functions, order of basis, differences, partial sums, partial sums of reciprocals, sum of reciprocals) about the centered polygonal numbers.

Centered polygonal numbers formulae and values

A-number	Centered V -gonal numbers	Formula  c P  (2) V(n)  =  V  Tn + 1	n = 0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	# of primes for n in [0 .. 19]
A005448  (n + 1)	Centered trigonal numbers (Centered triangular numbers)	$3T_{n}+1\,$	1	4	10	19	31	46	64	85	109	136	166	199	235	274	316	361	409	460	514	571	6
A001844  (n)	Centered tetragonal numbers (Centered square numbers)	$4T_{n}+1\,$	1	5	13	25	41	61	85	113	145	181	221	265	313	365	421	481	545	613	685	761	10
A005891  (n)	Centered pentagonal numbers	$5T_{n}+1\,$	1	6	16	31	51	76	106	141	181	226	276	331	391	456	526	601	681	766	856	951	4
A003215  (n)	Centered hexagonal numbers (Hex numbers)	$6T_{n}+1\,$	1	7	19	37	61	91	127	169	217	271	331	397	469	547	631	721	817	919	1027	1141	11
A069099  (n + 1)	Centered heptagonal numbers	$7T_{n}+1\,$	1	8	22	43	71	106	148	197	253	316	386	463	547	638	736	841	953	1072	1198	1331	6
A016754  (n)	Centered octagonal numbers	$8T_{n}+1\,$ $(2n+1)^{2}\,$	1	9	25	49	81	121	169	225	289	361	441	529	625	729	841	961	1089	1225	1369	1521	0
A060544  (n + 1)	Centered nonagonal numbers	$9T_{n}+1\,$ $T_{[3n+1]}\,$	1	10	28	55	91	136	190	253	325	406	496	595	703	820	946	1081	1225	1378	1540	1711	0
A062786  (n + 1)	Centered decagonal numbers	$10T_{n}+1\,$	1	11	31	61	101	151	211	281	361	451	551	661	781	911	1051	1201	1361	1531	1711	1901	14
A069125  (n + 1)	Centered hendecagonal numbers	$11T_{n}+1\,$	1	12	34	67	111	166	232	309	397	496	606	727	859	1002	1156	1321	1497	1684	1882	2091	5
A003154  (n + 1)	Centered dodecagonal numbers	$12T_{n}+1\,$	1	13	37	73	121	181	253	337	433	541	661	793	937	1093	1261	1441	1633	1837	2053	2281	12
A069126  (n + 1)	Centered tridecagonal numbers	$13T_{n}+1\,$	1	14	40	79	131	196	274	365	469	586	716	859	1015	1184	1366	1561	1769	1990	2224	2471	3
A069127  (n + 1)	Centered tetradecagonal numbers	$14T_{n}+1\,$	1	15	43	85	141	211	295	393	505	631	771	925	1093	1275	1471	1681	1905	2143	2395	2661	6
A069128  (n + 1)	Centered pentadecagonal numbers	$15T_{n}+1\,$	1	16	46	91	151	226	316	421	541	676	826	991	1171	1366	1576	1801	2041	2296	2566	2851	7
A069129  (n + 1)	Centered hexadecagonal numbers	$16T_{n}+1\,$	1	17	49	97	161	241	337	449	577	721	881	1057	1249	1457	1681	1921	2177	2449	2737	3041	9
A069130  (n + 1)	Centered heptadecagonal numbers	$17T_{n}+1\,$	1	18	52	103	171	256	358	477	613	766	936	1123	1327	1548	1786	2041	2313	2602	2908	3231	4
A069131  (n + 1)	Centered octadecagonal numbers	$18T_{n}+1\,$	1	19	55	109	181	271	379	505	649	811	991	1189	1405	1639	1891	2161	2449	2755	3079	3421	9
A069132  (n + 1)	Centered nonadecagonal numbers	$19T_{n}+1\,$	1	20	58	115	191	286	400	533	685	856	1046	1255	1483	1730	1996	2281	2585	2908	3250	3611	3
A069133  (n + 1)	Centered icosagonal numbers	$20T_{n}+1\,$	1	21	61	121	201	301	421	561	721	901	1101	1321	1561	1821	2101	2401	2721	3061	3421	3801	4

Three-dimensional figurate numbers

Pyramidal numbers

The pyramidal numbers

Y  (3) V(n)

model

(V  −  1)

-gonal (the number of sides of the polygonal base of a pyramid being equal to its number of vertices

V

minus one for the apex vertex) pyramids in which each horizontal layer corresponds to a regular convex polygonal number, thus:

Y  (3) V(n)  ≡    n ∑   i   = 0    P  (2) [V −1] (i).

The general formula for the pyramidal numbers is:

Y  (3) V(n)  =  ([V − 1] − 2) n 3 + 3 n 2 − ([V − 1] − 5) n 6  =  n 6 {([V − 1] − 2) n 2 + 3 n − ([V − 1] − 5)}.

Prime numbers (shown in bold) may appear only for

Y  (3) V(2) = V + 1

since

Y  (3) V(n), n   ≥   3,

is obviously composite from the formula.

See Pyramidal numbers (and Category:Pyramidal numbers) for more information (e.g. more formulae, recurrence equations, generating functions, order of basis, differences, partial sums, partial sums of reciprocals, sum of reciprocals) about the pyramidal numbers.

Pyramidal numbers formulae and values

A-number	(V  −  1) -gonal pyramidal numbers	Formula Y  (3) V(n)	n = 0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
A000292  (n)	Trigonal pyramidal numbers (Tetrahedral numbers)	${\frac {n^{3}+3n^{2}+2n}{6}}\,$ $P_{3+1}^{(3)}(n)\,$	0	1	4	10	20	35	56	84	120	165	220	286	364	455	560	680	816	969	1140	1330	1540
A000330  (n)	Tetragonal pyramidal numbers (Square pyramidal numbers)	${\frac {2n^{3}+3n^{2}+n}{6}}\,$ ${\frac {n(2n+1)(n+1)}{6}}\,$ ${\frac {1}{4}}{\binom {2n+2}{3}}\,$	0	1	5	14	30	55	91	140	204	285	385	506	650	819	1015	1240	1496	1785	2109	2470	2870
A002411  (n)	Pentagonal pyramidal numbers	${\frac {n^{3}+n^{2}}{2}}\,$	0	1	6	18	40	75	126	196	288	405	550	726	936	1183	1470	1800	2176	2601	3078	3610	4200
A002412  (n)	Hexagonal pyramidal numbers	${\frac {4n^{3}+3n^{2}-n}{6}}\,$	0	1	7	22	50	95	161	252	372	525	715	946	1222	1547	1925	2360	2856	3417	4047	4750	5530
A002413  (n)	Heptagonal pyramidal numbers	${\frac {5n^{3}+3n^{2}-2n}{6}}\,$	0	1	8	26	60	115	196	308	456	645	880	1166	1508	1911	2380	2920	3536	4233	5016	5890	6860
A002414  (n)	Octagonal pyramidal numbers	${\frac {2n^{3}+n^{2}-n}{2}}\,$	0	1	9	30	70	135	231	364	540	765	1045	1386	1794	2275	2835	3480	4216	5049	5985	7030	8190
A007584  (n)	Nonagonal pyramidal numbers	${\frac {7n^{3}+3n^{2}-4n}{6}}\,$	0	1	10	34	80	155	266	420	624	885	1210	1606	2080	2639	3290	4040	4896	5865	6954	8170	9520
A007585  (n)	Decagonal pyramidal numbers	${\frac {8n^{3}+3n^{2}-5n}{6}}\,$	0	1	11	38	90	175	301	476	708	1005	1375	1826	2366	3003	3745	4600	5576	6681	7923	9310	10850
A007586  (n)	Hendecagonal pyramidal numbers	${\frac {3n^{3}+n^{2}-2n}{2}}\,$	0	1	12	42	100	195	336	532	792	1125	1540	2046	2652	3367	4200	5160	6256	7497	8892	10450	12180
A007587  (n)	Dodecagonal pyramidal numbers	${\frac {10n^{3}+3n^{2}-7n}{6}}\,$	0	1	13	46	110	215	371	588	876	1245	1705	2266	2938	3731	4655	5720	6936	8313	9861	11590	13510
A050441  (n)	Tridecagonal pyramidal numbers	${\frac {11n^{3}+3n^{2}-8n}{6}}\,$	0	1	14	50	120	235	406	644	960	1365	1870	2486	3224	4095	5110	6280	7616	9129	10830	12730	14840
A172073  (n)	Tetradecagonal pyramidal numbers	${\frac {4n^{3}+n^{2}-3n}{2}}\,$	0	1	15	54	130	255	441	700	1044	1485	2035	2706	3510	4459	5565	6840	8296	9945	11799	13870	16170
A??????  (n)	Pentadecagonal pyramidal numbers	${\frac {13n^{3}+3n^{2}-10n}{6}}\,$	0	1	16	58	140	275	476	756	1128	1605	2200	2926	3796	4823	6020	7400	8976	10761	12768	15010	17500
A172076  (n)	Hexadecagonal pyramidal numbers	${\frac {14n^{3}+3n^{2}-11n}{6}}\,$	0	1	17	62	150	295	511	812	1212	1725	2365	3146	4082	5187	6475	7960	9656	11577	13737	16150	18830
A??????  (n)	Heptadecagonal pyramidal numbers	${\frac {5n^{3}+n^{2}-4n}{2}}\,$	0	1	18	66	160	315	546	868	1296	1845	2530	3366	4368	5551	6930	8520	10336	12393	14706	17290	20160
A172078  (n)	Octadecagonal pyramidal numbers	${\frac {16n^{3}+3n^{2}-13n}{6}}\,$	0	1	19	70	170	335	581	924	1380	1965	2695	3586	4654	5915	7385	9080	11016	13209	15675	18430	21490
A??????  (n)	Nonadecagonal pyramidal numbers	${\frac {17n^{3}+3n^{2}-14n}{6}}\,$	0	1	20	74	180	355	616	980	1464	2085	2860	3806	4940	6279	7840	9640	11696	14025	16644	19570	22820
A172082  (n)	Icosagonal pyramidal numbers	${\frac {6n^{3}+n^{2}-5n}{2}}\,$	0	1	21	78	190	375	651	1036	1548	2205	3025	4026	5226	6643	8295	10200	12376	14841	17613	20710	24150

Centered pyramidal numbers

The centered pyramidal numbers, i.e. (centered polygons) pyramidal numbers are a family of sequences of 3-dimensional nonregular polytope numbers (among the 3-dimensional figurate numbers) formed by adding the first

[N0  −  1]

positive centered polygonal numbers with constant number of sides

[N0  −  1]

, where

N0

is the number of vertices (including the apex vertex) of the polygonal base pyramid. The term centered pyramid numbers, i.e. (centered squares) pyramidal numbers, is often used to refer to the centered square pyramidal numbers, i.e. (centered squares) pyramidal numbers, having a polygonal base with four sides. The centered pyramidal numbers, i.e. (centered polygons) pyramidal numbers, are a generalization of the centered pyramid numbers, i.e. (centered squares) pyramidal numbers, where the base is a regular convex polygon with any number of sides

[N0  −  1]   ≥   3

. Centered pyramidal numbers, i.e. (centered polygons) pyramidal numbers, may also be generalized to higher dimensions as centered hyperpyramidal numbers, i.e. (centered polygons) hyperpyramidal numbers. While the (centered polygons) pyramidal numbers are pyramidal stacks of centered polygons, the generated figures are not globally centered. Thus the (centered polygons) pyramidal numbers do not belong to the category of globally centered figurate numbers (which start with the globally central dot, giving value 1, for

n = 0

), they belong to the category of globally noncentered figurate numbers (which equals 0 for

n = 0

) and start with the initial dot, giving value 1, for

n = 1

). It would be less confusing if the centered pyramidal numbers were called (centered polygons) pyramidal numbers. Note that although the triangular pyramidal numbers are tetrahedral numbers, the (centered triangles) pyramidal numbers are NOT the (globally centered) centered tetrahedral numbers. The centered pyramidal numbers

‘c’Y  (3) V(n)

model

(V  −  1)

-gonal (the number of sides of the polygonal base of a pyramid being equal to the number of vertices

V

minus one for the apex vertex) pyramids in which each horizontal layer corresponds to a centered polygonal number, thus

‘c’Y  (3) V(n)  ≡    n ∑   i   = 0     c P  (2) [V −1] (i).

The general formula for centered pyramidal numbers is

‘c’Y  (3) V(n)  =  [V − 1] n 3 − ([V − 1] − 6) n 6  =  [V − 1] (n − 1) n (n + 1) 6 + n  =  [V − 1] (  n + 13  )  + n.

Observe that the

n

th centered hexagonal pyramidal number is equal to the

n

th cube. Also the

n

th centered square pyramidal number is equal to the

n

th octahedral number, which is interesting since the

n

th octahedral number is the

n

th square dipyramidal number which is the sum of the

n

th square pyramidal number and the

(n  −  1)

th square pyramidal number. While the centered pyramidal numbers are pyramidal stacks of centered polygons, the generated figures are not globally centered. Thus the centered pyramidal numbers do not belong to the category of globally centered figurate numbers (which start with the globally central dot, giving value 1, for

n = 0

), they belong to the category of globally noncentered figurate numbers (which equals 0 for

n = 0

and start with the initial dot, giving value 1, for

n = 1

). It would be less confusing if the centered pyramidal numbers were called centered polygon pyramidal numbers. Note that although the triangular pyramidal numbers are tetrahedral numbers, the (centered polygons) centered triangular pyramidal numbers are NOT the (globally centered) centered tetrahedral numbers.

See Centered pyramidal numbers (and Category:Centered pyramidal numbers) for more information (e.g. more formulae, recurrence equations, generating functions, order of basis, differences, partial sums, partial sums of reciprocals, sum of reciprocals) about the centered pyramidal numbers.

Centered pyramidal numbers formulae and values

A-number	Centered (V  −  1) -gonal pyramidal numbers	Formula  ‘c’Y  (3) V(n)	n = 0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
A006003  (n)	Centered trigonal pyramidal numbers (Centered triangular pyramidal numbers)	${\frac {n^{3}+n}{2}}\,$	0	1	5	15	34	65	111	175	260	369	505	671	870	1105	1379	1695	2056	2465	2925	3439	4010
A005900  (n)	Centered tetragonal pyramidal numbers (Centered square pyramidal numbers)	${\frac {2n^{3}+n}{3}}\,$ $P^{(3)}(2\cdot 3,n)\,$ Octahedral numbers	0	1	6	19	44	85	146	231	344	489	670	891	1156	1469	1834	2255	2736	3281	3894	4579	5340
A004068  (n)	Centered pentagonal pyramidal numbers	${\frac {5n^{3}+n}{6}}\,$	0	1	7	23	54	105	181	287	428	609	835	1111	1442	1833	2289	2815	3416	4097	4863	5719	6670
A000578  (n)	Centered hexagonal pyramidal numbers (Hex pyramidal numbers)	$n^{3}\,$ $P^{(3)}(2^{3},n)\,$ Cubes	0	1	8	27	64	125	216	343	512	729	1000	1331	1728	2197	2744	3375	4096	4913	5832	6859	8000
A004126  (n)	Centered heptagonal pyramidal numbers	${\frac {7n^{3}-n}{6}}\,$	0	1	9	31	74	145	251	399	596	849	1165	1551	2014	2561	3199	3935	4776	5729	6801	7999	9330
A000447  (n)	Centered octagonal pyramidal numbers	${\frac {4n^{3}-n}{3}}\,$	0	1	10	35	84	165	286	455	680	969	1330	1771	2300	2925	3654	4495	5456	6545	7770	9139	10660
A004188  (n)	Centered nonagonal pyramidal numbers	${\frac {3n^{3}-n}{2}}\,$	0	1	11	39	94	185	321	511	764	1089	1495	1991	2586	3289	4109	5055	6136	7361	8739	10279	11990
A004466  (n)	Centered decagonal pyramidal numbers	${\frac {5n^{3}-2n}{3}}\,$	0	1	12	43	104	205	356	567	848	1209	1660	2211	2872	3653	4564	5615	6816	8177	9708	11419	13320
A004467  (n)	Centered hendecagonal pyramidal numbers	${\frac {11n^{3}-5n}{6}}\,$	0	1	13	47	114	225	391	623	932	1329	1825	2431	3158	4017	5019	6175	7496	8993	10677	12559	14650
A007588  (n)	Centered dodecagonal pyramidal numbers	$2n^{3}-n\,$ $P^{(3)}(2\cdot 3,n)+8\,P^{(3)}(3+1,n-1)\,$ Stellated octahedron number  (Stella octangula number)	0	1	14	51	124	245	426	679	1016	1449	1990	2651	3444	4381	5474	6735	8176	9809	11646	13699	15980
A062025  (n)	Centered tridecagonal pyramidal numbers	${\frac {13n^{3}-7n}{6}}\,$	0	1	15	55	134	265	461	735	1100	1569	2155	2871	3730	4745	5929	7295	8856	10625	12615	14839	17310
A063521  (n)	Centered tetradecagonal pyramidal numbers	${\frac {7n^{3}-4n}{3}}\,$	0	1	16	59	144	285	496	791	1184	1689	2320	3091	4016	5109	6384	7855	9536	11441	13584	15979	18640
A063522  (n)	Centered pentadecagonal pyramidal numbers	${\frac {5n^{3}-3n}{2}}\,$	0	1	17	63	154	305	531	847	1268	1809	2485	3311	4302	5473	6839	8415	10216	12257	14553	17119	19970
A063523  (n)	Centered hexadecagonal pyramidal numbers	${\frac {8n^{3}-5n}{3}}\,$	0	1	18	67	164	325	566	903	1352	1929	2650	3531	4588	5837	7294	8975	10896	13073	15522	18259	21300
A??????  (n)	Centered heptadecagonal pyramidal numbers	${\frac {17n^{3}-11n}{6}}\,$	0	1	19	71	174	345	601	959	1436	2049	2815	3751	4874	6201	7749	9535	11576	13889	16491	19399	22630
A??????  (n)	Centered octadecagonal pyramidal numbers	$3n^{3}-2n\,$	0	1	20	75	184	365	636	1015	1520	2169	2980	3971	5160	6565	8204	10095	12256	14705	17460	20539	23960
A??????  (n)	Centered nonadecagonal pyramidal numbers	${\frac {19n^{3}-13n}{6}}\,$	0	1	21	79	194	385	671	1071	1604	2289	3145	4191	5446	6929	8659	10655	12936	15521	18429	21679	25290
A??????  (n)	Centered icosagonal pyramidal numbers	${\frac {10n^{3}-7n}{3}}\,$	0	1	22	83	204	405	706	1127	1688	2409	3310	4411	5732	7293	9114	11215	13616	16337	19398	22819	26620

See also

• Classifications of figurate numbers

External links

• Table of Figurate Numbers, Sorted, Through 10,000, compiled by Magic Dragon Multimedia, 2004.
• Table of Polytope Numbers, Sorted, Through 1,000,000, compiled by Magic Dragon Multimedia, 2004.