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Centered polygonal numbers

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Centered hexagonal numbers
     [1]
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

Contents

Formulae

The nth centered N0-gonal number, where n = 0 gives the central dot, is given by the formula:[2]

\,_cP^{(2)}_{N_0}(n) = N_0\ P^{(2)}_{3}(n) + 1 = N_0\ T_{n} + 1 = N_0 \binom{n+1}{2} + 1 = N_0 {{n(n+1)}\over{2}} + 1,\,

where \scriptstyle P^{(2)}_{3}(n) = T_n\, is the nth triangular number.

Schläfli-Poincaré (convex) polytope formula

Schläfli-Poincaré generalization of the Descartes-Euler (convex) polyhedral formula.[3]

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

\,_cP^{(2)}_{N_0}(n) = \,_cP^{(2)}_{N_0}(n-1) + N_0\ n,\,

with initial condition

\,_cP^{(2)}_{N_0}(0) = 1.\,

Generating function

G_{\{\,_cP^{(2)}_{N_0}(n)\}}(x) = {{x^2+(N_0-2)x+1}\over{(1-x)^3}}\,

Order of basis

The order of basis of centered
N0
-gonal numbers is:
g_{ \{ {\,}_{c}P^{(2)}_{N_0} \} } = \ ?,\quad 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[4]) 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[5]), 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
{nd | n = 0, 1, 2, }
of
d
th powers forms a basis of order
g (d)
. The Hilbert-Waring problem[6] 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,[7] 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

\,_cP^{(2)}_{N_0}(n) - \,_cP^{(2)}_{N_0}(n-1) = N_0\ n = N_0\ P^{(1)}_{1}(n)\,

Partial sums

\sum_{n=0}^m {\,_cP^{(2)}_{N_0}(n)} = N_0 \frac{m(m+1)(m+2)}{6} + m = N_0 \binom{m+2}{3} + m = N_0\ P^{(3)}_{4}(m) + m\,

Partial sums of reciprocals

\sum_{n=0}^m \frac{1}{\,_cP^{(2)}_{N_0}(n)} = ...\,

Sum of reciprocals

\sum_{n=0}^{\infty} \frac{1}{\,_cP^{(2)}_{N_0}(n)} = \frac{2\pi}{N_0 \sqrt{1-\frac{8}{N_0}}} \tan{\bigg( \frac{\pi}{2} \sqrt{1-\frac{8}{N_0}} \bigg)},\ N_0 \neq 8,\,
= \sum_{n=0}^{\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

\,_cP^{(2)}_{N_0}(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 A??????
26 Centered icosihexagonal 26T_n+1\,

13n(n+1)+1\,

1 27 79 157 261 391 547 729 937 1171 1431 1717 A??????
27 Centered icosiheptagonal 27T_n+1\,

27n(n+1)/2+1\,

1 28 82 163 271 406 568 757 973 1216 1486 1783 A??????
28 Centered icosioctagonal 28T_n+1\,

14n(n+1)+1\,

1 29 85 169 281 421 589 785 1009 1261 1541 1849 A??????
29 Centered icosinonagonal 29T_n+1\,

29n(n+1)/2+1\,

1 30 88 175 291 436 610 813 1045 1306 1596 1915 A??????
30 Centered triacontagonal 30T_n+1\,

15n(n+1)+1\,

1 31 91 181 301 451 631 841 1081 1351 1651 1981 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_{\{\,_cP^{(2)}_{N_0}(n)\}}(x) =\,


{{x^2+(N_0-2)x+1}\over{(1-x)^3}}\,

Order

of basis

g_{\{\,_cP^{(2)}_{N_0}\}}\,

Differences

\,_cP^{(2)}_{N_0}(n) - \,

\,_cP^{(2)}_{N_0}(n-1) =\,


N_0\ n\,

Partial sums

\sum_{n=0}^m {\,_cP^{(2)}_{N_0}(n)} =


N_0 \binom{m+2}{3} + m\,


N_0\ P^{(3)}_{4}(m) + m\,

Partial sums of reciprocals

\sum_{n=0}^m {1\over{\,_cP^{(2)}_{N_0}(n)}} =

Sum of Reciprocals[8]

\sum_{n=0}^\infty{1\over{\,_cP^{(2)}_{N_0}(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 \,_cP^{(2)}_{N_0}(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

Notes

  1. Author of the plots: Stefan Friedrich Birkner, License: Creative Commons Attribution-ShareAlike 3.0 Unported.
  2. Where \scriptstyle \,_cP^{(d)}_{N_0}(n)\, is the d-dimensional centered regular convex polytope number with N0 vertices.
  3. Weisstein, Eric W., Polyhedral Formula, From MathWorld--A Wolfram Web Resource.
  4. Weisstein, Eric W., Lagrange's Four-Square Theorem, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/LagrangesFour-SquareTheorem.html]
  5. Weisstein, Eric W., Fermat's Polygonal Number Theorem, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/FermatsPolygonalNumberTheorem.html]
  6. Weisstein, Eric W., Waring's Problem, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/WaringsProblem.html]
  7. Weisstein, Eric W., Fifteen Theorem, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/FifteenTheorem.html]
  8. 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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