Centered polygonal number

The centered polygonal numbers are a class of series of figurate numbers, each formed by a central dot, surrounded 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, so starting from the second polygonal layer each layer of a centered k-gonal number contains k more points than the previous layer.

Examples

Each element in the sequence is a multiple of the previous triangular number plus 1. This can be formalized by the equation ${\frac {ax(x+1)}{2}}+1$ where a is the number of sides of the polygon, and x is the sequence number, starting with zero for the initial 1. For example, the centered square numbers are four times the triangular numbers plus 1, or equivalently ${\frac {4x(x+1)}{2}}+1$ .

These series consist of the

centered triangular numbers 1, 4, 10, 19, 31, 46, 64, 85, 109, 136, 166, 199, ... (OEIS: A005448)
centered square numbers 1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 265, ... (OEIS: A001844)
centered pentagonal numbers 1, 6, 16, 31, 51, 76, 106, 141, 181, 226, 276, 331, ... (OEIS: A005891)
centered hexagonal numbers 1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, ... (OEIS: A003215), which are exactly the difference of consecutive cubes, i.e. x³ − (x − 1)³
centered heptagonal numbers 1, 8, 22, 43, 71, 106, 148, 197, 253, 316, 386, 463, ... (OEIS: A069099)
centered octagonal numbers 1, 9, 25, 49, 81, 121, 169, 225, 289, 361, 441, 529, ... (OEIS: A016754), which are exactly the odd squares
centered nonagonal numbers 1, 10, 28, 55, 91, 136, 190, 253, 325, 406, 496, 595, ... (OEIS: A060544), which include all even perfect numbers except 6
centered decagonal numbers 1, 11, 31, 61, 101, 151, 211, 281, 361, 451, 551, 661, ... (OEIS: A062786)
centered hendecagonal numbers 1, 12, 34, 67, 111, 166, 232, 309, 397, 496, 606, 727, ... (OEIS: A069125)
centered dodecagonal numbers 1, 13, 37, 73, 121, 181, 253, 337, 433, 541, 661, 793, ... (OEIS: A003154), which are also the star numbers

and so on.

The following diagrams show a few examples of centered polygonal numbers and their geometric construction. Compare these diagrams with the diagrams in Polygonal number.

centered triangular number	centered square number	centered pentagonal number	centered hexagonal number

Centered square numbers

1		5		13		25

Centered hexagonal numbers

1		7		19		37

Formula

As can be seen in the above diagrams, the nth centered k-gonal number can be obtained by placing k copies of the (n−1)th triangular number around a central point; therefore, the nth centered k-gonal number can be mathematically represented by

C_{k,n}={\frac {kn}{2}}(n-1)+1.

The difference of the n-th and the (n+1)-th consecutive centered k-gonal numbers is k(2n+1).

The n-th centered k-gonal number is equal to the n-th regular k-gonal number plus (n-1)².

Just as is the case with regular polygonal numbers, the first centered k-gonal number is 1. Thus, for any k, 1 is both k-gonal and centered k-gonal. The next number to be both k-gonal and centered k-gonal can be found using the formula:

{\frac {k^{2}}{2}}(k-1)+1

which tells us that 10 is both triangular and centered triangular, 25 is both square and centered square, etc.

Whereas a prime number p cannot be a polygonal number (except the trivial case, i.e. each p is the second p-gonal number), many centered polygonal numbers are primes. In fact, if k ≥ 3, k ≠ 8, k ≠ 9, then there are infinitely many centered k-gonal numbers which are primes (assuming the Bunyakovsky conjecture). (Since all centered octagonal numbers are also square numbers, and all centered nonagonal numbers are also triangular numbers (and not equal to 3), thus both of them cannot be prime numbers)

Sum of Reciprocals

The sum of reciprocals for the centered k-gonal numbers is^[1]

{\frac {2\pi }{k{\sqrt {1-{\frac {8}{k}}}}}}\tan \left({\frac {\pi }{2}}{\sqrt {1-{\frac {8}{k}}}}\right)

, if k ≠ 8

{\frac {\pi ^{2}}{8}}

, if k = 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	$9T_{n}+1\,$ $9n(n+1)/2+1\,$ $t_{3n+1}\,$ ${\binom {3n+2}{2}}$	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 $G_{\{\,_{c}P_{N_{0}}^{(2)}(n)\}}(x)=\,$ ${{x^{2}+(N_{0}-2)x+1} \over {(1-x)^{3}}}\,$	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 $\sum _{n=0}^{m}{\,_{c}P_{N_{0}}^{(2)}(n)}=$ $N_{0}{\binom {m+2}{3}}+m\,$ $N_{0}\ P_{4}^{(3)}(m)+m\,$	Partial sums of reciprocals $\sum _{n=0}^{m}{1 \over {\,_{c}P_{N_{0}}^{(2)}(n)}}=$	Sum of Reciprocals^[2] $\sum _{n=0}^{\infty }{1 \over {\,_{c}P_{N_{0}}^{(2)}(n)}}=$ $\scriptstyle {{\frac {2\pi }{N_{0}{\sqrt {1-{\tfrac {8}{N_{0}}}}}}}\tan {{\big (}{\frac {\pi }{2}}{\sqrt {1-{\tfrac {8}{N_{0}}}}}{\big )}}},\,$ $\scriptstyle N_{0}\neq 8,\,$ ${\frac {\pi ^{2}}{8}},\ N_{0}=8.\,$
3	Centered triangular	${x^{2}+x+1} \over {(1-x)^{3}}\,$	$\,$	$3n\,$	$3{\binom {m+2}{3}}+m\,$	$\,$	$\,$
4	Centered square	${x^{2}+2x+1} \over {(1-x)^{3}}\,$ ${(x+1)^{2}} \over {(1-x)^{3}}\,$	$\,$	$4n\,$	$4{\binom {m+2}{3}}+m\,$	$\,$	${\frac {\pi }{2}}\tanh {\bigg (}{\frac {\pi }{2}}{\bigg )}\,$
5	Centered pentagonal	${x^{2}+3x+1} \over {(1-x)^{3}}\,$	$\,$	$5n\,$	$5{\binom {m+2}{3}}+m\,$	$\,$	$\,$
6	Centered hexagonal	${x^{2}+4x+1} \over {(1-x)^{3}}\,$	$\,$	$6n\,$	$6{\binom {m+2}{3}}+m\,$	$\,$	${\frac {\pi }{\sqrt {3}}}\tanh {\bigg (}{\frac {\pi }{2{\sqrt {3}}}}{\bigg )}\,$
7	Centered heptagonal	${x^{2}+5x+1} \over {(1-x)^{3}}\,$	$\,$	$7n\,$	$7{\binom {m+2}{3}}+m\,$	$\,$	${\frac {2\pi }{\sqrt {7}}}\tanh {\bigg (}{\frac {\pi }{2{\sqrt {7}}}}{\bigg )}\,$
8	Centered octagonal	${x^{2}+6x+1} \over {(1-x)^{3}}\,$	$\,$	$8n\,$	$8{\binom {m+2}{3}}+m\,$	$\,$	${\frac {\pi ^{2}}{8}}\,$
9	Centered nonagonal	${x^{2}+7x+1} \over {(1-x)^{3}}\,$	$\,$	$9n\,$	$9{\binom {m+2}{3}}+m\,$	$\,$	${\frac {2\pi }{3}}\tan {\bigg (}{\frac {\pi }{6}}{\bigg )}\,$
10	Centered decagonal	${x^{2}+8x+1} \over {(1-x)^{3}}\,$	$\,$	$10n\,$	$10{\binom {m+2}{3}}+m\,$	$\,$	${\frac {\pi }{\sqrt {5}}}\tan {\bigg (}{\frac {\pi }{2{\sqrt {5}}}}{\bigg )}\,$
11	Centered hendecagonal	${x^{2}+9x+1} \over {(1-x)^{3}}\,$	$\,$	$11n\,$	$11{\binom {m+2}{3}}+m\,$	$\,$	$\,$
12	Centered dodecagonal	${x^{2}+10x+1} \over {(1-x)^{3}}\,$	$\,$	$12n\,$	$12{\binom {m+2}{3}}+m\,$	$\,$	${\frac {\pi }{2{\sqrt {3}}}}\tan {\bigg (}{\frac {\pi }{2{\sqrt {3}}}}{\bigg )}\,$
13	Centered tridecagonal	${x^{2}+11x+1} \over {(1-x)^{3}}\,$	$\,$	$13n\,$	$13{\binom {m+2}{3}}+m\,$	$\,$	$\,$
14	Centered tetradecagonal	${x^{2}+12x+1} \over {(1-x)^{3}}\,$	$\,$	$14n\,$	$14{\binom {m+2}{3}}+m\,$	$\,$	$\,$
15	Centered pentadecagonal	${x^{2}+13x+1} \over {(1-x)^{3}}\,$	$\,$	$15n\,$	$15{\binom {m+2}{3}}+m\,$	$\,$	$\,$
16	Centered hexadecagonal	${x^{2}+14x+1} \over {(1-x)^{3}}\,$	$\,$	$16n\,$	$16{\binom {m+2}{3}}+m\,$	$\,$	${\frac {\pi }{4{\sqrt {2}}}}\tan {\bigg (}{\frac {\pi }{2{\sqrt {2}}}}{\bigg )}\,$
17	Centered heptadecagonal	${x^{2}+15x+1} \over {(1-x)^{3}}\,$	$\,$	$17n\,$	$17{\binom {m+2}{3}}+m\,$	$\,$	$\,$
18	Centered octadecagonal	${x^{2}+16x+1} \over {(1-x)^{3}}\,$	$\,$	$18n\,$	$18{\binom {m+2}{3}}+m\,$	$\,$	$\,$
19	Centered nonadecagonal	${x^{2}+17x+1} \over {(1-x)^{3}}\,$	$\,$	$19n\,$	$19{\binom {m+2}{3}}+m\,$	$\,$	$\,$
20	Centered icosagonal	${x^{2}+18x+1} \over {(1-x)^{3}}\,$	$\,$	$20n\,$	$20{\binom {m+2}{3}}+m\,$	$\,$	$\,$
21	Centered icosihenagonal	${x^{2}+19x+1} \over {(1-x)^{3}}\,$	$\,$	$21n\,$	$21{\binom {m+2}{3}}+m\,$	$\,$	$\,$
22	Centered icosidigonal	${x^{2}+20x+1} \over {(1-x)^{3}}\,$	$\,$	$22n\,$	$22{\binom {m+2}{3}}+m\,$	$\,$	$\,$
23	Centered icositrigonal	${x^{2}+21x+1} \over {(1-x)^{3}}\,$	$\,$	$23n\,$	$23{\binom {m+2}{3}}+m\,$	$\,$	$\,$
24	Centered icositetragonal	${x^{2}+22x+1} \over {(1-x)^{3}}\,$	$\,$	$24n\,$	$24{\binom {m+2}{3}}+m\,$	$\,$	${\frac {\pi }{4{\sqrt {6}}}}\tan {\bigg (}{\frac {\pi }{\sqrt {6}}}{\bigg )}\,$
25	Centered icosipentagonal	${x^{2}+23x+1} \over {(1-x)^{3}}\,$	$\,$	$25n\,$	$25{\binom {m+2}{3}}+m\,$	$\,$	$\,$
26	Centered icosihexagonal	${x^{2}+24x+1} \over {(1-x)^{3}}\,$	$\,$	$26n\,$	$26{\binom {m+2}{3}}+m\,$	$\,$	$\,$
27	Centered icosiheptagonal	${x^{2}+25x+1} \over {(1-x)^{3}}\,$	$\,$	$27n\,$	$27{\binom {m+2}{3}}+m\,$	$\,$	$\,$
28	Centered icosioctagonal	${x^{2}+26x+1} \over {(1-x)^{3}}\,$	$\,$	$28n\,$	$28{\binom {m+2}{3}}+m\,$	$\,$	$\,$
29	Centered icosinonagonal	${x^{2}+27x+1} \over {(1-x)^{3}}\,$	$\,$	$29n\,$	$29{\binom {m+2}{3}}+m\,$	$\,$	$\,$
30	Centered triacontagonal	${x^{2}+28x+1} \over {(1-x)^{3}}\,$	$\,$	$30n\,$	$30{\binom {m+2}{3}}+m\,$	$\,$	${\frac {\pi }{6{\sqrt {5}}}}\tan {\bigg (}{\frac {\pi }{\sqrt {5}}}{\bigg )}\,$

Table of sequences

**Centered polygonal numbers sequences**
N₀	$\,_{c}P_{N_{0}}^{(2)}(n),\ n\geq 0$ 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, ...}

References

^ centered polygonal numbers in OEIS wiki, content "Table of related formulae and values"
^ Downey, Lawrence M., Ong, Boon W., and Sellers, James A., Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, 2008.

Neil Sloane & Simon Plouffe (1995). The Encyclopedia of Integer Sequences. San Diego: Academic Press.: Fig. M3826
Weisstein, Eric W. "Centered polygonal number". MathWorld.
F. Tapson (1999). The Oxford Mathematics Study Dictionary (2nd ed.). Oxford University Press. pp. 88–89. ISBN 0-19-914-567-9.

[1] tered polygonal numbers in OEIS wiki, content "Table of related formulae and values"

[2] Downey, Lawrence M., Ong, Boon W., and Sellers, James A., Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, 2008.

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