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Centered polygonal numbers
[1] | |||
Centered triangular numbers | Centered square numbers | Centered pentagonal numbers | Centered hexagonal numbers (Hex numbers) |
n = 0 |
N0 |
V |
N1 |
N0 |
E |
N0 |
N0 |
All figurate numbers are accessible via this structured menu: Classifications of figurate numbers
Contents
- 1 Formulae
- 2 Schläfli-Poincaré (convex) polytope formula
- 3 Recurrence relation
- 4 Generating function
- 5 Order of basis
- 6 Differences
- 7 Partial sums
- 8 Partial sums of reciprocals
- 9 Sum of reciprocals
- 10 Table of formulae and values
- 11 Table of related formulae and values
- 12 Table of sequences
- 13 See also
- 14 Notes
- 15 External links
Formulae
The nth centered N0-gonal number, where n = 0 gives the central dot, is given by the formula:[2]
where 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:
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
with initial condition
Generating function
Order of basis
The order of basis of centeredN0 |
k |
k |
k |
k |
A |
g |
g |
g |
A |
{n 2 | n = 0, 1, 2, …} |
k ≥ 3 |
{P (k, n) | n = 0, 1, 2, …} |
k |
k |
k |
k |
g (d) |
g (d) |
d |
{n d | n = 0, 1, 2, …} |
d |
g (d) |
g (d) |
d ≥ 2 |
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
Partial sums
Partial sums of reciprocals
Sum of reciprocals
Table of formulae and values
Centered polygonal numbers associated with constructible polygons (Cf. A003401) (with straightedge and compass) are named in bold.
N0 | Name | Formulae
|
n = 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | OEIS
number |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3 | Centered triangular |
|
1 | 4 | 10 | 19 | 31 | 46 | 64 | 85 | 109 | 136 | 166 | 199 | A005448(n+1) |
4 | Centered square |
|
1 | 5 | 13 | 25 | 41 | 61 | 85 | 113 | 145 | 181 | 221 | 265 | A001844(n) |
5 | Centered pentagonal |
|
1 | 6 | 16 | 31 | 51 | 76 | 106 | 141 | 181 | 226 | 276 | 331 | A005891(n) |
6 | Centered hexagonal |
|
1 | 7 | 19 | 37 | 61 | 91 | 127 | 169 | 217 | 271 | 331 | 397 | A003215(n) |
7 | Centered heptagonal |
|
1 | 8 | 22 | 43 | 71 | 106 | 148 | 197 | 253 | 316 | 386 | 463 | A069099(n+1) |
8 | Centered octagonal |
|
1 | 9 | 25 | 49 | 81 | 121 | 169 | 225 | 289 | 361 | 441 | 529 | A016754(n) |
9 | Centered nonagonal |
|
1 | 10 | 28 | 55 | 91 | 136 | 190 | 253 | 325 | 406 | 496 | 595 | A060544(n+1) |
10 | Centered decagonal |
|
1 | 11 | 31 | 61 | 101 | 151 | 211 | 281 | 361 | 451 | 551 | 661 | A062786(n+1) |
11 | Centered hendecagonal |
|
1 | 12 | 34 | 67 | 111 | 166 | 232 | 309 | 397 | 496 | 606 | 727 | A069125(n+1) |
12 | Centered dodecagonal |
|
1 | 13 | 37 | 73 | 121 | 181 | 253 | 337 | 433 | 541 | 661 | 793 | A003154(n+1) |
13 | Centered tridecagonal |
|
1 | 14 | 40 | 79 | 131 | 196 | 274 | 365 | 469 | 586 | 716 | 859 | A069126(n+1) |
14 | Centered tetradecagonal |
|
1 | 15 | 43 | 85 | 141 | 211 | 295 | 393 | 505 | 631 | 771 | 925 | A069127(n+1) |
15 | Centered pentadecagonal |
|
1 | 16 | 46 | 91 | 151 | 226 | 316 | 421 | 541 | 676 | 826 | 991 | A069128(n+1) |
16 | Centered hexadecagonal |
|
1 | 17 | 49 | 97 | 161 | 241 | 337 | 449 | 577 | 721 | 881 | 1057 | A069129(n+1) |
17 | Centered heptadecagonal |
|
1 | 18 | 52 | 103 | 171 | 256 | 358 | 477 | 613 | 766 | 936 | 1123 | A069130(n+1) |
18 | Centered octadecagonal |
|
1 | 19 | 55 | 109 | 181 | 271 | 379 | 505 | 649 | 811 | 991 | 1189 | A069131(n+1) |
19 | Centered nonadecagonal |
|
1 | 20 | 58 | 115 | 191 | 286 | 400 | 533 | 685 | 856 | 1046 | 1255 | A069132(n+1) |
20 | Centered icosagonal |
|
1 | 21 | 61 | 121 | 201 | 301 | 421 | 561 | 721 | 901 | 1101 | 1321 | A069133(n+1) |
21 | Centered icosihenagonal |
|
1 | 22 | 64 | 127 | 211 | 316 | 442 | 589 | 757 | 946 | 1156 | 1387 | A069178(n+1) |
22 | Centered icosidigonal |
|
1 | 23 | 67 | 133 | 221 | 331 | 463 | 617 | 793 | 991 | 1211 | 1453 | A069173(n+1) |
23 | Centered icositrigonal |
|
1 | 24 | 70 | 139 | 231 | 346 | 484 | 645 | 829 | 1036 | 1266 | 1519 | A069174(n+1) |
24 | Centered icositetragonal |
|
1 | 25 | 73 | 145 | 241 | 361 | 505 | 673 | 865 | 1081 | 1321 | 1585 | A069190(n+1) |
25 | Centered icosipentagonal |
|
1 | 26 | 76 | 151 | 251 | 376 | 526 | 701 | 901 | 1126 | 1376 | 1651 | A?????? |
26 | Centered icosihexagonal |
|
1 | 27 | 79 | 157 | 261 | 391 | 547 | 729 | 937 | 1171 | 1431 | 1717 | A?????? |
27 | Centered icosiheptagonal |
|
1 | 28 | 82 | 163 | 271 | 406 | 568 | 757 | 973 | 1216 | 1486 | 1783 | A?????? |
28 | Centered icosioctagonal |
|
1 | 29 | 85 | 169 | 281 | 421 | 589 | 785 | 1009 | 1261 | 1541 | 1849 | A?????? |
29 | Centered icosinonagonal |
|
1 | 30 | 88 | 175 | 291 | 436 | 610 | 813 | 1045 | 1306 | 1596 | 1915 | A?????? |
30 | Centered triacontagonal |
|
1 | 31 | 91 | 181 | 301 | 451 | 631 | 841 | 1081 | 1351 | 1651 | 1981 | A?????? |
Centered polygonal numbers associated with constructible polygons (Cf. A003401) (with straightedge and compass) are named in bold.
N0 | Name | Generating
function
|
Order
of basis
|
Differences
|
Partial sums
|
Partial sums of reciprocals
|
Sum of Reciprocals[8]
|
---|---|---|---|---|---|---|---|
3 | Centered triangular | ||||||
4 | Centered square |
|
|||||
5 | Centered pentagonal | ||||||
6 | Centered hexagonal | ||||||
7 | Centered heptagonal | ||||||
8 | Centered octagonal | ||||||
9 | Centered nonagonal | ||||||
10 | Centered decagonal | ||||||
11 | Centered hendecagonal | ||||||
12 | Centered dodecagonal | ||||||
13 | Centered tridecagonal | ||||||
14 | Centered tetradecagonal | ||||||
15 | Centered pentadecagonal | ||||||
16 | Centered hexadecagonal | ||||||
17 | Centered heptadecagonal | ||||||
18 | Centered octadecagonal | ||||||
19 | Centered nonadecagonal | ||||||
20 | Centered icosagonal | ||||||
21 | Centered icosihenagonal | ||||||
22 | Centered icosidigonal | ||||||
23 | Centered icositrigonal | ||||||
24 | Centered icositetragonal | ||||||
25 | Centered icosipentagonal | ||||||
26 | Centered icosihexagonal | ||||||
27 | Centered icosiheptagonal | ||||||
28 | Centered icosioctagonal | ||||||
29 | Centered icosinonagonal | ||||||
30 | Centered triacontagonal |
Table of sequences
N0 | 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
Notes
- ↑ Author of the plots: Stefan Friedrich Birkner, License: Creative Commons Attribution-ShareAlike 3.0 Unported.
- ↑ Where is the d-dimensional centered regular convex polytope number with N0 vertices.
- ↑ Weisstein, Eric W., Polyhedral Formula, From MathWorld--A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Lagrange's Four-Square Theorem, from MathWorld—A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Fermat's Polygonal Number Theorem, from MathWorld—A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Waring's Problem, from MathWorld—A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Fifteen Theorem, from MathWorld—A Wolfram Web Resource.
- ↑ Downey, Lawrence M., Ong, Boon W., and Sellers, James A., Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, 2008.
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.