Centered polygonal numbers

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

The centered polygonal numbers are a family of sequences of 2-dimensional regular polytope numbers (among the 2-dimensional figurate numbers), each formed by a central dot (for  

n = 0

), surrounded by polygonal layers with a constant number  

N0

of 0-dimensional elements (or vertices  

V

 ), thus a constant number  

N1

equal to  

N0

of 1-dimensional elements (or edges  

E

). Each side of a polygonal layer contains one more dot than a side in the previous layer, so starting from the second polygonal layer each layer of a centered  

N0

-gonal number contains  

N0

more points than the previous layer.

All figurate numbers are accessible via this structured menu: Classifications of figurate numbers

Formulae

The n^th centered N₀-gonal number, where n = 0 gives the central dot, is given by the formula:

${}_c{P}_{N_0}^{(2)}(n)=N_0{P}_3^{(2)}(n)+1=N_0{T}_n+1=N_0(\genfrac{}{}{0ex}{}{n+1}{2})+1=N_0\frac{n(n+1)}{2}+1,$

where $P_3^{(2)}(n)=T_n$ is the n^th triangular number.

Schläfli-Poincaré (convex) polytope formula

Schläfli-Poincaré generalization of the Descartes-Euler (convex) polyhedral formula.

For nondegenerate 2-dimensional regular convex polygons:

$\sum _{i=0}^1(-1{)}^i{N}_i=N_0-N_1=V-E=0,$

where N₀ is the number of 0-dimensional elements (vertices V,) N₁ is the number of 1-dimensional elements (edges E) of the convex polygon.

Recurrence relation

${}_c{P}_{N_0}^{(2)}(n)={}_c{P}_{N_0}^{(2)}(n-1)+N_{0}n,$

with initial condition

${}_c{P}_{N_0}^{(2)}(0)=1.$

Generating function

$G_{\{{}_c{P}_{N_0}^{(2)}(n)\}}(x)=\frac{x^2+(N_0-2)x+1}{(1-x{)}^3}$

Order of basis

The order of basis of centered  

N0

-gonal numbers is:

$g_{\{{}_c{P}_{N_0}^{(2)}\}}=?,N_0\ge 3.$

In 1638, Fermat proposed that every positive integer is a sum of at most three triangular numbers, four square numbers, five pentagonal numbers, and  

k

 

k

-gonal numbers. Fermat claimed to have a proof of this result, although Fermat's proof has never been found. Joseph Louis Lagrange proved the square case (known as the four squares theorem) in 1770 and Gauss proved the triangular case in 1796. In 1813, Cauchy finally proved the horizontal generalization that every nonnegative integer can be written as a sum of  

k

 

k

-gonal numbers (known as the polygonal number theorem), while a vertical (higher dimensional) generalization has also been made (known as the Hilbert-Waring problem). A nonempty subset  

A

of nonnegative integers is called a basis of order  

g

if  

g

is the minimum number with the property that every nonnegative integer can be written as a sum of  

g

elements in  

A

. Lagrange’s sum of four squares can be restated as the set  

{n 2 | n = 0, 1, 2, …}

of nonnegative squares forms a basis of order 4. Theorem (Cauchy) For every  

k   ≥   3

, the set  

{P (k, n) | n = 0, 1, 2, …}

of  

k

-gon numbers forms a basis of order  

k

, i.e. every nonnegative integer can be written as a sum of  

k

 

k

-gon numbers. We note that polygonal numbers are two dimensional analogues of squares. Obviously, cubes, fourth powers, fifth powers, ... are higher dimensional analogues of squares. In 1770, Waring stated without proof that every nonnegative integer can be written as a sum of 4 squares, 9 cubes, 19 fourth powers, and so on. In 1909, Hilbert proved that there is a finite number  

g (d)

such that every nonnegative integer is a sum of  

g (d)

 

d

 th powers, i.e. the set  

{n d | n = 0, 1, 2, …}

of  

d

 th powers forms a basis of order  

g (d)

. The Hilbert-Waring problem is concerned with the study of  

g (d)

for  

d   ≥   2

. This problem was one of the most important research topics in additive number theory in last 90 years, and it is still a very active area of research.

In 1997, Conway et al. proved a theorem, called the fifteen theorem, which states that, if a positive definite quadratic form with integer matrix entries represents all natural numbers up to 15, then it represents all natural numbers. This theorem contains Lagrange's four-square theorem, since every number up to 15 is the sum of at most four squares.

Differences

${}_c{P}_{N_0}^{(2)}(n)-{}_c{P}_{N_0}^{(2)}(n-1)=N_{0}n=N_0{P}_1^{(1)}(n)$

Partial sums

$\sum _{n=0}^m{}_c{P}_{N_0}^{(2)}(n)=N_0\frac{m(m+1)(m+2)}{6}+m=N_0(\genfrac{}{}{0ex}{}{m+2}{3})+m=N_0{P}_4^{(3)}(m)+m$

Partial sums of reciprocals

$\sum _{n=0}^m\frac{1}{{}_c{P}_{N_0}^{(2)}(n)}=...$

Sum of reciprocals

$\sum _{n=0}^{\mathrm{\infty }}\frac{1}{{}_c{P}_{N_0}^{(2)}(n)}=\frac{2\pi }{N_0\sqrt{1-\frac{8}{N_0}}}\tan \left(\frac{\pi }{2}\sqrt{1-\frac{8}{N_0}}\right),N_0\ne 8,$

$=\sum _{n=0}^{\mathrm{\infty }}\frac{1}{(2n+1{)}^2}=\frac{{\pi }^2}{8},N_0=8.$

Table of formulae and values

Centered polygonal numbers associated with constructible polygons (Cf. A003401) (with straightedge and compass) are named in bold.

Centered polygonal numbers formulae and values

N0	Name	Formulae ${}_c{P}_{N_0}^{(2)}(n)$	n = 0	1	2	3	4	5	6	7	8	9	10	11	OEIS number
3	Centered triangular	$3T_n+1$ $3n(n+1)/2+1$	1	4	10	19	31	46	64	85	109	136	166	199	A005448(n+1)
4	Centered square	$4T_n+1$ $2n(n+1)+1$ $n^2+(n+1{)}^2$	1	5	13	25	41	61	85	113	145	181	221	265	A001844(n)
5	Centered pentagonal	$5T_n+1$ $5n(n+1)/2+1$	1	6	16	31	51	76	106	141	181	226	276	331	A005891(n)
6	Centered hexagonal Hex numbers	$6T_n+1$ $3n(n+1)+1$	1	7	19	37	61	91	127	169	217	271	331	397	A003215(n)
7	Centered heptagonal	$7T_n+1$ $7n(n+1)/2+1$	1	8	22	43	71	106	148	197	253	316	386	463	A069099(n+1)
8	Centered octagonal	$8T_n+1$ $4n(n+1)+1$ $(2n+1{)}^2$ Odd squares	1	9	25	49	81	121	169	225	289	361	441	529	A016754(n)
9	Centered nonagonal	$9T_n+1$ $9n(n+1)/2+1$ $t_{3n+1}$ $(\genfrac{}{}{0ex}{}{3n+2}{2})$	1	10	28	55	91	136	190	253	325	406	496	595	A060544(n+1)
10	Centered decagonal	$10T_n+1$ $5n(n+1)+1$	1	11	31	61	101	151	211	281	361	451	551	661	A062786(n+1)
11	Centered hendecagonal	$11T_n+1$ $11n(n+1)/2+1$	1	12	34	67	111	166	232	309	397	496	606	727	A069125(n+1)
12	Centered dodecagonal	$12T_n+1$ $6n(n+1)+1$	1	13	37	73	121	181	253	337	433	541	661	793	A003154(n+1)
13	Centered tridecagonal	$13T_n+1$ $13n(n+1)/2+1$	1	14	40	79	131	196	274	365	469	586	716	859	A069126(n+1)
14	Centered tetradecagonal	$14T_n+1$ $7n(n+1)+1$	1	15	43	85	141	211	295	393	505	631	771	925	A069127(n+1)
15	Centered pentadecagonal	$15T_n+1$ $15n(n+1)/2+1$	1	16	46	91	151	226	316	421	541	676	826	991	A069128(n+1)
16	Centered hexadecagonal	$16T_n+1$ $8n(n+1)+1$	1	17	49	97	161	241	337	449	577	721	881	1057	A069129(n+1)
17	Centered heptadecagonal	$17T_n+1$ $17n(n+1)/2+1$	1	18	52	103	171	256	358	477	613	766	936	1123	A069130(n+1)
18	Centered octadecagonal	$18T_n+1$ $9n(n+1)+1$	1	19	55	109	181	271	379	505	649	811	991	1189	A069131(n+1)
19	Centered nonadecagonal	$19T_n+1$ $19n(n+1)/2+1$	1	20	58	115	191	286	400	533	685	856	1046	1255	A069132(n+1)
20	Centered icosagonal	$20T_n+1$ $10n(n+1)+1$	1	21	61	121	201	301	421	561	721	901	1101	1321	A069133(n+1)
21	Centered icosihenagonal	$21T_n+1$ $21n(n+1)/2+1$	1	22	64	127	211	316	442	589	757	946	1156	1387	A069178(n+1)
22	Centered icosidigonal	$22T_n+1$ $11n(n+1)+1$	1	23	67	133	221	331	463	617	793	991	1211	1453	A069173(n+1)
23	Centered icositrigonal	$23T_n+1$ $23n(n+1)/2+1$	1	24	70	139	231	346	484	645	829	1036	1266	1519	A069174(n+1)
24	Centered icositetragonal	$24T_n+1$ $12n(n+1)+1$	1	25	73	145	241	361	505	673	865	1081	1321	1585	A069190(n+1)
25	Centered icosipentagonal	$25T_n+1$ $25n(n+1)/2+1$	1	26	76	151	251	376	526	701	901	1126	1376	1651	OEIS:A??????
26	Centered icosihexagonal	$26T_n+1$ $13n(n+1)+1$	1	27	79	157	261	391	547	729	937	1171	1431	1717	OEIS:A??????
27	Centered icosiheptagonal	$27T_n+1$ $27n(n+1)/2+1$	1	28	82	163	271	406	568	757	973	1216	1486	1783	OEIS:A??????
28	Centered icosioctagonal	$28T_n+1$ $14n(n+1)+1$	1	29	85	169	281	421	589	785	1009	1261	1541	1849	OEIS:A??????
29	Centered icosinonagonal	$29T_n+1$ $29n(n+1)/2+1$	1	30	88	175	291	436	610	813	1045	1306	1596	1915	OEIS:A??????
30	Centered triacontagonal	$30T_n+1$ $15n(n+1)+1$	1	31	91	181	301	451	631	841	1081	1351	1651	1981	OEIS:A??????

Table of related formulae and values

Centered polygonal numbers associated with constructible polygons (Cf. A003401) (with straightedge and compass) are named in bold.

Centered polygonal numbers related formulae and values

N0	Name	Generating function $G_{\{{}_c{P}_{N_0}^{(2)}(n)\}}(x)=$ $\frac{x^2+(N_0-2)x+1}{(1-x{)}^3}$	Order of basis $g_{\{{}_c{P}_{N_0}^{(2)}\}}$	Differences ${}_c{P}_{N_0}^{(2)}(n)-$ ${}_c{P}_{N_0}^{(2)}(n-1)=$ $N_{0}n$	Partial sums $\sum _{n=0}^m{}_c{P}_{N_0}^{(2)}(n)=$ $N_0(\genfrac{}{}{0ex}{}{m+2}{3})+m$ $N_0{P}_4^{(3)}(m)+m$	Partial sums of reciprocals $\sum _{n=0}^m\frac{1}{{}_c{P}_{N_0}^{(2)}(n)}=$	Sum of Reciprocals $\sum _{n=0}^{\mathrm{\infty }}\frac{1}{{}_c{P}_{N_0}^{(2)}(n)}=$ $\frac{2\pi }{N_0\sqrt{1-\frac{8}{N_0}}}\tan \left(\frac{\pi }{2}\sqrt{1-\frac{8}{N_0}}\right),$ $N_0\ne 8,$ $\frac{{\pi }^2}{8},N_0=8.$
3	Centered triangular	$\frac{x^2+x+1}{(1-x{)}^3}$	$$	$3n$	$3(\genfrac{}{}{0ex}{}{m+2}{3})+m$	$$	$$
4	Centered square	$\frac{x^2+2x+1}{(1-x{)}^3}$ $\frac{(x+1{)}^2}{(1-x{)}^3}$	$$	$4n$	$4(\genfrac{}{}{0ex}{}{m+2}{3})+m$	$$	$\frac{\pi }{2}\tanh \left(\frac{\pi }{2}\right)$
5	Centered pentagonal	$\frac{x^2+3x+1}{(1-x{)}^3}$	$$	$5n$	$5(\genfrac{}{}{0ex}{}{m+2}{3})+m$	$$	$$
6	Centered hexagonal	$\frac{x^2+4x+1}{(1-x{)}^3}$	$$	$6n$	$6(\genfrac{}{}{0ex}{}{m+2}{3})+m$	$$	$\frac{\pi }{\sqrt{3}}\tanh \left(\frac{\pi }{2\sqrt{3}}\right)$
7	Centered heptagonal	$\frac{x^2+5x+1}{(1-x{)}^3}$	$$	$7n$	$7(\genfrac{}{}{0ex}{}{m+2}{3})+m$	$$	$\frac{2\pi }{\sqrt{7}}\tanh \left(\frac{\pi }{2\sqrt{7}}\right)$
8	Centered octagonal	$\frac{x^2+6x+1}{(1-x{)}^3}$	$$	$8n$	$8(\genfrac{}{}{0ex}{}{m+2}{3})+m$	$$	$\frac{{\pi }^2}{8}$
9	Centered nonagonal	$\frac{x^2+7x+1}{(1-x{)}^3}$	$$	$9n$	$9(\genfrac{}{}{0ex}{}{m+2}{3})+m$	$$	$\frac{2\pi }{3}\tan \left(\frac{\pi }{6}\right)$
10	Centered decagonal	$\frac{x^2+8x+1}{(1-x{)}^3}$	$$	$10n$	$10(\genfrac{}{}{0ex}{}{m+2}{3})+m$	$$	$\frac{\pi }{\sqrt{5}}\tan \left(\frac{\pi }{2\sqrt{5}}\right)$
11	Centered hendecagonal	$\frac{x^2+9x+1}{(1-x{)}^3}$	$$	$11n$	$11(\genfrac{}{}{0ex}{}{m+2}{3})+m$	$$	$$
12	Centered dodecagonal	$\frac{x^2+10x+1}{(1-x{)}^3}$	$$	$12n$	$12(\genfrac{}{}{0ex}{}{m+2}{3})+m$	$$	$\frac{\pi }{2\sqrt{3}}\tan \left(\frac{\pi }{2\sqrt{3}}\right)$
13	Centered tridecagonal	$\frac{x^2+11x+1}{(1-x{)}^3}$	$$	$13n$	$13(\genfrac{}{}{0ex}{}{m+2}{3})+m$	$$	$$
14	Centered tetradecagonal	$\frac{x^2+12x+1}{(1-x{)}^3}$	$$	$14n$	$14(\genfrac{}{}{0ex}{}{m+2}{3})+m$	$$	$$
15	Centered pentadecagonal	$\frac{x^2+13x+1}{(1-x{)}^3}$	$$	$15n$	$15(\genfrac{}{}{0ex}{}{m+2}{3})+m$	$$	$$
16	Centered hexadecagonal	$\frac{x^2+14x+1}{(1-x{)}^3}$	$$	$16n$	$16(\genfrac{}{}{0ex}{}{m+2}{3})+m$	$$	$\frac{\pi }{4\sqrt{2}}\tan \left(\frac{\pi }{2\sqrt{2}}\right)$
17	Centered heptadecagonal	$\frac{x^2+15x+1}{(1-x{)}^3}$	$$	$17n$	$17(\genfrac{}{}{0ex}{}{m+2}{3})+m$	$$	$$
18	Centered octadecagonal	$\frac{x^2+16x+1}{(1-x{)}^3}$	$$	$18n$	$18(\genfrac{}{}{0ex}{}{m+2}{3})+m$	$$	$$
19	Centered nonadecagonal	$\frac{x^2+17x+1}{(1-x{)}^3}$	$$	$19n$	$19(\genfrac{}{}{0ex}{}{m+2}{3})+m$	$$	$$
20	Centered icosagonal	$\frac{x^2+18x+1}{(1-x{)}^3}$	$$	$20n$	$20(\genfrac{}{}{0ex}{}{m+2}{3})+m$	$$	$$
21	Centered icosihenagonal	$\frac{x^2+19x+1}{(1-x{)}^3}$	$$	$21n$	$21(\genfrac{}{}{0ex}{}{m+2}{3})+m$	$$	$$
22	Centered icosidigonal	$\frac{x^2+20x+1}{(1-x{)}^3}$	$$	$22n$	$22(\genfrac{}{}{0ex}{}{m+2}{3})+m$	$$	$$
23	Centered icositrigonal	$\frac{x^2+21x+1}{(1-x{)}^3}$	$$	$23n$	$23(\genfrac{}{}{0ex}{}{m+2}{3})+m$	$$	$$
24	Centered icositetragonal	$\frac{x^2+22x+1}{(1-x{)}^3}$	$$	$24n$	$24(\genfrac{}{}{0ex}{}{m+2}{3})+m$	$$	$\frac{\pi }{4\sqrt{6}}\tan \left(\frac{\pi }{\sqrt{6}}\right)$
25	Centered icosipentagonal	$\frac{x^2+23x+1}{(1-x{)}^3}$	$$	$25n$	$25(\genfrac{}{}{0ex}{}{m+2}{3})+m$	$$	$$
26	Centered icosihexagonal	$\frac{x^2+24x+1}{(1-x{)}^3}$	$$	$26n$	$26(\genfrac{}{}{0ex}{}{m+2}{3})+m$	$$	$$
27	Centered icosiheptagonal	$\frac{x^2+25x+1}{(1-x{)}^3}$	$$	$27n$	$27(\genfrac{}{}{0ex}{}{m+2}{3})+m$	$$	$$
28	Centered icosioctagonal	$\frac{x^2+26x+1}{(1-x{)}^3}$	$$	$28n$	$28(\genfrac{}{}{0ex}{}{m+2}{3})+m$	$$	$$
29	Centered icosinonagonal	$\frac{x^2+27x+1}{(1-x{)}^3}$	$$	$29n$	$29(\genfrac{}{}{0ex}{}{m+2}{3})+m$	$$	$$
30	Centered triacontagonal	$\frac{x^2+28x+1}{(1-x{)}^3}$	$$	$30n$	$30(\genfrac{}{}{0ex}{}{m+2}{3})+m$	$$	$\frac{\pi }{6\sqrt{5}}\tan \left(\frac{\pi }{\sqrt{5}}\right)$

Table of sequences

Centered polygonal numbers sequences

N0	${}_c{P}_{N_0}^{(2)}(n),n\ge 0$ sequences
3	{1, 4, 10, 19, 31, 46, 64, 85, 109, 136, 166, 199, 235, 274, 316, 361, 409, 460, 514, 571, 631, 694, 760, 829, 901, 976, 1054, 1135, 1219, 1306, 1396, 1489, 1585, 1684, ...}
4	{1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 265, 313, 365, 421, 481, 545, 613, 685, 761, 841, 925, 1013, 1105, 1201, 1301, 1405, 1513, 1625, 1741, 1861, 1985, 2113, ...}
5	{1, 6, 16, 31, 51, 76, 106, 141, 181, 226, 276, 331, 391, 456, 526, 601, 681, 766, 856, 951, 1051, 1156, 1266, 1381, 1501, 1626, 1756, 1891, 2031, 2176, 2326, 2481, 2641, ...}
6	{1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, 469, 547, 631, 721, 817, 919, 1027, 1141, 1261, 1387, 1519, 1657, 1801, 1951, 2107, 2269, 2437, 2611, 2791, 2977, 3169, ...}
7	{1, 8, 22, 43, 71, 106, 148, 197, 253, 316, 386, 463, 547, 638, 736, 841, 953, 1072, 1198, 1331, 1471, 1618, 1772, 1933, 2101, 2276, 2458, 2647, 2843, 3046, 3256, 3473, ...}
8	{1, 9, 25, 49, 81, 121, 169, 225, 289, 361, 441, 529, 625, 729, 841, 961, 1089, 1225, 1369, 1521, 1681, 1849, 2025, 2209, 2401, 2601, 2809, 3025, 3249, 3481, 3721, 3969, ...}
9	{1, 10, 28, 55, 91, 136, 190, 253, 325, 406, 496, 595, 703, 820, 946, 1081, 1225, 1378, 1540, 1711, 1891, 2080, 2278, 2485, 2701, 2926, 3160, 3403, 3655, 3916, 4186, 4465, ...}
10	{1, 11, 31, 61, 101, 151, 211, 281, 361, 451, 551, 661, 781, 911, 1051, 1201, 1361, 1531, 1711, 1901, 2101, 2311, 2531, 2761, 3001, 3251, 3511, 3781, 4061, 4351, 4651, ...}
11	{1, 12, 34, 67, 111, 166, 232, 309, 397, 496, 606, 727, 859, 1002, 1156, 1321, 1497, 1684, 1882, 2091, 2311, 2542, 2784, 3037, 3301, 3576, 3862, 4159, 4467, 4786, ...}
12	{1, 13, 37, 73, 121, 181, 253, 337, 433, 541, 661, 793, 937, 1093, 1261, 1441, 1633, 1837, 2053, 2281, 2521, 2773, 3037, 3313, 3601, 3901, 4213, 4537, 4873, 5221, 5581, ...}
13	{1, 14, 40, 79, 131, 196, 274, 365, 469, 586, 716, 859, 1015, 1184, 1366, 1561, 1769, 1990, 2224, 2471, 2731, 3004, 3290, 3589, 3901, 4226, 4564, 4915, 5279, 5656, 6046, ...}
14	{1, 15, 43, 85, 141, 211, 295, 393, 505, 631, 771, 925, 1093, 1275, 1471, 1681, 1905, 2143, 2395, 2661, 2941, 3235, 3543, 3865, 4201, 4551, 4915, 5293, 5685, 6091, 6511, ...}
15	{1, 16, 46, 91, 151, 226, 316, 421, 541, 676, 826, 991, 1171, 1366, 1576, 1801, 2041, 2296, 2566, 2851, 3151, 3466, 3796, 4141, 4501, 4876, 5266, 5671, 6091, 6526, 6976, ...}
16	{1, 17, 49, 97, 161, 241, 337, 449, 577, 721, 881, 1057, 1249, 1457, 1681, 1921, 2177, 2449, 2737, 3041, 3361, 3697, 4049, 4417, 4801, 5201, 5617, 6049, 6497, 6961, 7441, ...}
17	{1, 18, 52, 103, 171, 256, 358, 477, 613, 766, 936, 1123, 1327, 1548, 1786, 2041, 2313, 2602, 2908, 3231, 3571, 3928, 4302, 4693, 5101, 5526, 5968, 6427, 6903, 7396, 7906, ...}
18	{1, 19, 55, 109, 181, 271, 379, 505, 649, 811, 991, 1189, 1405, 1639, 1891, 2161, 2449, 2755, 3079, 3421, 3781, 4159, 4555, 4969, 5401, 5851, 6319, 6805, 7309, 7831, 8371, ...}
19	{1, 20, 58, 115, 191, 286, 400, 533, 685, 856, 1046, 1255, 1483, 1730, 1996, 2281, 2585, 2908, 3250, 3611, 3991, 4390, 4808, 5245, 5701, 6176, 6670, 7183, 7715, 8266, 8836, ...}
20	{1, 21, 61, 121, 201, 301, 421, 561, 721, 901, 1101, 1321, 1561, 1821, 2101, 2401, 2721, 3061, 3421, 3801, 4201, 4621, 5061, 5521, 6001, 6501, 7021, 7561, 8121, 8701, 9301, ...}
21	{1, 22, 64, 127, 211, 316, 442, 589, 757, 946, 1156, 1387, 1639, 1912, 2206, 2521, 2857, 3214, 3592, 3991, 4411, 4852, 5314, 5797, 6301, 6826, 7372, 7939, 8527, 9136, 9766, ...}
22	{1, 23, 67, 133, 221, 331, 463, 617, 793, 991, 1211, 1453, 1717, 2003, 2311, 2641, 2993, 3367, 3763, 4181, 4621, 5083, 5567, 6073, 6601, 7151, 7723, 8317, 8933, 9571, 10231, ...}
23	{1, 24, 70, 139, 231, 346, 484, 645, 829, 1036, 1266, 1519, 1795, 2094, 2416, 2761, 3129, 3520, 3934, 4371, 4831, 5314, 5820, 6349, 6901, 7476, 8074, 8695, 9339, 10006, ...}
24	{1, 25, 73, 145, 241, 361, 505, 673, 865, 1081, 1321, 1585, 1873, 2185, 2521, 2881, 3265, 3673, 4105, 4561, 5041, 5545, 6073, 6625, 7201, 7801, 8425, 9073, 9745, 10441, ...}
25	{1, 26, 76, 151, 251, 376, 526, 701, 901, 1126, 1376, 1651, 1951, 2276, 2626, 3001, 3401, 3826, 4276, 4751, 5251, 5776, 6326, 6901, 7501, 8126, 8776, 9451, 10151, 10876, ...}
26	{1, 27, 79, 157, 261, 391, 547, 729, 937, 1171, 1431, 1717, 2029, 2367, 2731, 3121, 3537, 3979, 4447, 4941, 5461, 6007, 6579, 7177, 7801, 8451, 9127, 9829, 10557, 11311, ...}
27	{1, 28, 82, 163, 271, 406, 568, 757, 973, 1216, 1486, 1783, 2107, 2458, 2836, 3241, 3673, 4132, 4618, 5131, 5671, 6238, 6832, 7453, 8101, 8776, 9478, 10207, 10963, 11746, ...}
28	{1, 29, 85, 169, 281, 421, 589, 785, 1009, 1261, 1541, 1849, 2185, 2549, 2941, 3361, 3809, 4285, 4789, 5321, 5881, 6469, 7085, 7729, 8401, 9101, 9829, 10585, 11369, 12181, ...}
29	{1, 30, 88, 175, 291, 436, 610, 813, 1045, 1306, 1596, 1915, 2263, 2640, 3046, 3481, 3945, 4438, 4960, 5511, 6091, 6700, 7338, 8005, 8701, 9426, 10180, 10963, 11775, 12616, ...}
30	{1, 31, 91, 181, 301, 451, 631, 841, 1081, 1351, 1651, 1981, 2341, 2731, 3151, 3601, 4081, 4591, 5131, 5701, 6301, 6931, 7591, 8281, 9001, 9751, 10531, 11341, 12181, 13051, ...}

See also

Polygonal numbers

External links

• S. Plouffe, Approximations de Séries Génératrices et Quelques Conjectures, Dissertation, Université du Québec à Montréal, 1992.
• S. Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.
• Herbert S. Wilf, generatingfunctionology, 1994.