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Polygonal numbers
From OeisWiki
^{[1]}  
Triangular numbers  Square numbers  Pentagonal numbers  Hexagonal numbers 
The polygonal numbers are the family of sequences of 2dimensional convex regular polytope numbers, made of n successive polygonal layers with a constant number N_{0} of 0dimensional elements (vertices V of the polygons), having n+1 dots for each edge (including both end vertices) of the n^{th} layer, n ≥ 1, with all layers sharing a common vertex (which corresponds to n = 0) and the two sides sharing that vertex.
The number N_{1} of 1dimensional elements (edges E of the polygons) equals the number N_{0} of 0dimensional elements (vertices V of the polygons.)
All figurate numbers are accessible via this structured menu: Classifications of figurate numbers
Formulae
The n^{th} N_{0}gonal number is given by the formulae:^{[2]}
where is the n^{th} N_{0}gonal gnomonic number, and where N_{0} is the number of 0dimensional elements (which are vertices V) of the polygons and is the n^{th} triangular number.
SchläfliPoincaré (convex) polytope formula
SchläfliPoincaré generalization of the DescartesEuler (convex) polyhedral formula.^{[3]}
For nondegenerate 2dimensional regular convex polygons:
where N_{0} is the number of 0dimensional elements (vertices V,) N_{1} is the number of 1dimensional elements (edges E) of the convex polygon.
Recurrence equation
with initial conditions
Generating function
Order of basis
The order of basis of N_{0}gonal numbers is:
The order of basis g for numbers of the form is k, since to represent the numbers in the congruence classes by adding numbers congruent to we need as many terms as the class number, for each congruence classes, e.g. for :
 numbers of form are expressible as 1 term of the form ;
 numbers of form are expressible as the sum of 2 terms of the form ;
 numbers of form are expressible as the sum of 3 terms of the form ;
 numbers of form are expressible as the sum of 4 terms of the form ;
 numbers of form are expressible as the sum of 5 terms of the form .
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 kpolygonal 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 kgon numbers (known as the polygonal number theorem,^{[4]}) while a vertical (higher dimensional) generalization has also been made (known as the HilbertWaring 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 of nonnegative squares forms a basis of order 4.
Theorem (Cauchy) For every , the set of kgon numbers forms a basis of order k, i.e. every nonnegative integer can be written as a sum of k kgon 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 such that every nonnegative integer is a sum of ^{th} powers, i.e. the set of ^{th} powers forms a basis of order . The HilbertWaring problem^{[5]} is concerned with the study of for . 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,^{[6]} 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 foursquare theorem, since every number up to 15 is the sum of at most four squares.
Differences
where is the n^{th} N_{0}gonal gnomonic number.
Partial sums
where is the m^{th} triangular number and is the m^{th} N_{0}gonal pyramidal number. ^{[7]}
Partial sums of reciprocals
For :
where is the m^{th} harmonic number,^{[8]} is the EulerMascheroni constant,^{[9]} and is the digamma function.^{[10]} ^{[11]}
For :
Sum of reciprocals
For :
For , the sum of reciprocals of the square numbers:
can be interpreted as , where is the probability that a random integer x is squarefree or that two random integers x and y are coprime, i.e. the random integer xy is squarefree.^{[12]}
Table of formulae and values
Polygonal numbers associated with constructible polygons (with straightedge and compass) (A003401) are named in bold.
Name  Formulae
 n = 0  1  2  3  4  5  6  7  8  9  10  11  12  Anumbers  

3  Triangular  0  1  3  6  10  15  21  28  36  45  55  66  78  A000217  
4  Square 
 0  1  4  9  16  25  36  49  64  81  100  121  144  A000290 
5  Pentagonal  0  1  5  12  22  35  51  70  92  117  145  176  210  A000326  
6  Hexagonal  0  1  6  15  28  45  66  91  120  153  190  231  276  A000384  
7  Heptagonal  0  1  7  18  34  55  81  112  148  189  235  286  342  A000566  
8  Octagonal  0  1  8  21  40  65  96  133  176  225  280  341  408  A000567  
9  9gonal  0  1  9  24  46  75  111  154  204  261  325  396  474  A001106  
10  10gonal  0  1  10  27  52  85  126  175  232  297  370  451  540  A001107  
11  11gonal  0  1  11  30  58  95  141  196  260  333  415  506  606  A051682  
12  12gonal  0  1  12  33  64  105  156  217  288  369  460  561  672  A051624  
13  13gonal  0  1  13  36  70  115  171  238  316  405  505  616  738  A051865  
14  14gonal  0  1  14  39  76  125  186  259  344  441  550  671  804  A051866  
15  15gonal  0  1  15  42  82  135  201  280  372  477  595  726  870  A051867  
16  16gonal  0  1  16  45  88  145  216  301  400  513  640  781  936  A051868  
17  17gonal  0  1  17  48  94  155  231  322  428  549  685  836  1002  A051869  
18  18gonal  0  1  18  51  100  165  246  343  456  585  730  891  1068  A051870  
19  19gonal  0  1  19  54  106  175  261  364  484  621  775  946  1134  A051871  
20  20gonal  0  1  20  57  112  185  276  385  512  657  820  1001  1200  A051872  
21  21gonal  0  1  21  60  118  195  291  406  540  693  865  1056  1266  A051873  
22  22gonal  0  1  22  63  124  205  306  427  568  729  910  1111  1332  A051874  
23  23gonal  0  1  23  66  130  215  321  448  596  765  955  1166  1398  A051875  
24  24gonal  0  1  24  69  136  225  336  469  624  801  1000  1221  1464  A051876  
25  25gonal  0  1  25  72  142  235  351  490  652  837  1045  1276  1530  A??????  
26  26gonal  0  1  26  75  148  245  366  511  680  873  1090  1331  1596  A??????  
27  27gonal  0  1  27  78  154  255  381  532  708  909  1135  1386  1662  A??????  
28  28gonal  0  1  28  81  160  265  396  553  736  945  1180  1441  1728  A??????  
29  29gonal  0  1  29  84  166  275  411  574  764  981  1225  1496  1794  A??????  
30  30gonal  0  1  30  87  172  285  426  595  792  1017  1270  1551  1860  A?????? 
Table of related formulae and values
N_{0} and N_{1} are the number of vertices (0dimensional) and edges (1dimensional) respectively, where the edges are the actual facets. The regular Platonic numbers are listed by increasing number N_{0} of vertices, which equals the number N_{1} of facets, or sides of the polygons.
Polygonal numbers associated with constructible polygons (with straightedge and compass) are named in bold.
N_{0}  Name
(N_{0}, N_{1}) Schläfli symbol^{[13]}  Generating function
 Order of basis
 Differences
 Partial sums
 Partial sums of reciprocals
 Sum of Reciprocals^{[14]}^{[15]}


3  Triangular
(3, 3) {3}  3 
 2(ψ(2) + γ)
 
4  Square
(4, 4) {4}  4 
 ^{[16]}  ^{[17]}
Base 10: A013661  
5  Pentagonal
(5, 5) {5}  5 


 
6  Hexagonal
(6, 6) {6}  6 
 2log(2)  
7  Heptagonal
(7, 7) {7}  7 
 
8  Octagonal
(8, 8) {8}  8 

 
9  9gonal
(9, 9) {9}  9 
 
10  10gonal
(10, 10) {10}  10 
 
11  11gonal
(11, 11) {11}  11 

 
12  12gonal
(12, 12) {12}  12 
 
13  13gonal
(13, 13) {13}  13 
 
14  14gonal
(14, 14) {14}  14 

 
15  15gonal
(15, 15) {15}  15 
 
16  16gonal
(16, 16) {16}  16 
 
17  17gonal
(17, 17) {17}  17 

 
18  18gonal
(18, 18) {18}  18 
 
19  19gonal
(19, 19) {19}  19 
 
20  20gonal
(20, 20) {20}  20 

 
21  21gonal
(21, 21) {21}  21 
 
22  22gonal
(22, 22) {22}  22 
 
23  23gonal
(23, 23) {23}  23 

 
24  24gonal
(24, 24) {24} 
 
25  25gonal
(25, 25) {25}  25 
 
26  26gonal
(26, 26) {26}  26 

 
27  27gonal
(27, 27) {27}  27 
 
28  28gonal
(28, 28) {28}  28 
 
29  29gonal
(29, 29) {29}  29 

 
30  30gonal
(30, 30) {30}  30 

Table of sequences
For OEIS sequence numbers, refer to table of formulae and values above.
N_{0}  sequences  Anumbers 

3  {0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, ...}  A?????? 
4  {0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, ...}  A?????? 
5  {0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, ...}  A?????? 
6  {0, 1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, 276, 325, 378, 435, 496, 561, 630, 703, 780, 861, 946, 1035, 1128, 1225, 1326, 1431, 1540, 1653, 1770, ...}  A?????? 
7  {0, 1, 7, 18, 34, 55, 81, 112, 148, 189, 235, 286, 342, 403, 469, 540, 616, 697, 783, 874, 970, 1071, 1177, 1288, 1404, 1525, 1651, 1782, 1918, 2059, ...}  A?????? 
8  {0, 1, 8, 21, 40, 65, 96, 133, 176, 225, 280, 341, 408, 481, 560, 645, 736, 833, 936, 1045, 1160, 1281, 1408, 1541, 1680, 1825, 1976, 2133, 2296, 2465, ...}  
9  {0, 1, 9, 24, 46, 75, 111, 154, 204, 261, 325, 396, 474, 559, 651, 750, 856, 969, 1089, 1216, 1350, 1491, 1639, 1794, 1956, 2125, 2301, 2484, 2674, 2871, ...}  A?????? 
10  {0, 1, 10, 27, 52, 85, 126, 175, 232, 297, 370, 451, 540, 637, 742, 855, 976, 1105, 1242, 1387, 1540, 1701, 1870, 2047, 2232, 2425, 2626, 2835, 3052, ...}  A?????? 
11  {0, 1, 11, 30, 58, 95, 141, 196, 260, 333, 415, 506, 606, 715, 833, 960, 1096, 1241, 1395, 1558, 1730, 1911, 2101, 2300, 2508, 2725, 2951, 3186, 3430, ...}  A?????? 
12  {0, 1, 12, 33, 64, 105, 156, 217, 288, 369, 460, 561, 672, 793, 924, 1065, 1216, 1377, 1548, 1729, 1920, 2121, 2332, 2553, 2784, 3025, 3276, 3537, 3808, ...}  A?????? 
13  {0, 1, 13, 36, 70, 115, 171, 238, 316, 405, 505, 616, 738, 871, 1015, 1170, 1336, 1513, 1701, 1900, 2110, 2331, 2563, 2806, 3060, 3325, 3601, 3888, 4186, ...}  A?????? 
14  {0, 1, 14, 39, 76, 125, 186, 259, 344, 441, 550, 671, 804, 949, 1106, 1275, 1456, 1649, 1854, 2071, 2300, 2541, 2794, 3059, 3336, 3625, 3926, 4239, 4564, ...}  A?????? 
15  {0, 1, 15, 42, 82, 135, 201, 280, 372, 477, 595, 726, 870, 1027, 1197, 1380, 1576, 1785, 2007, 2242, 2490, 2751, 3025, 3312, 3612, 3925, 4251, 4590, ...}  A?????? 
16  {0, 1, 16, 45, 88, 145, 216, 301, 400, 513, 640, 781, 936, 1105, 1288, 1485, 1696, 1921, 2160, 2413, 2680, 2961, 3256, 3565, 3888, 4225, 4576, 4941, ...}  A?????? 
17  {0, 1, 17, 48, 94, 155, 231, 322, 428, 549, 685, 836, 1002, 1183, 1379, 1590, 1816, 2057, 2313, 2584, 2870, 3171, 3487, 3818, 4164, 4525, 4901, 5292, ...}  A?????? 
18  {0, 1, 18, 51, 100, 165, 246, 343, 456, 585, 730, 891, 1068, 1261, 1470, 1695, 1936, 2193, 2466, 2755, 3060, 3381, 3718, 4071, 4440, 4825, 5226, 5643, ...}  A?????? 
19  {0, 1, 19, 54, 106, 175, 261, 364, 484, 621, 775, 946, 1134, 1339, 1561, 1800, 2056, 2329, 2619, 2926, 3250, 3591, 3949, 4324, 4716, 5125, 5551, 5994, ...}  A?????? 
20  {0, 1, 20, 57, 112, 185, 276, 385, 512, 657, 820, 1001, 1200, 1417, 1652, 1905, 2176, 2465, 2772, 3097, 3440, 3801, 4180, 4577, 4992, 5425, 5876, 6345, ...}  A?????? 
21  {0, 1, 21, 60, 118, 195, 291, 406, 540, 693, 865, 1056, 1266, 1495, 1743, 2010, 2296, 2601, 2925, 3268, 3630, 4011, 4411, 4830, 5268, 5725, 6201, 6696, ...}  A?????? 
22  {0, 1, 22, 63, 124, 205, 306, 427, 568, 729, 910, 1111, 1332, 1573, 1834, 2115, 2416, 2737, 3078, 3439, 3820, 4221, 4642, 5083, 5544, 6025, 6526, 7047, ...}  A?????? 
23  {0, 1, 23, 66, 130, 215, 321, 448, 596, 765, 955, 1166, 1398, 1651, 1925, 2220, 2536, 2873, 3231, 3610, 4010, 4431, 4873, 5336, 5820, 6325, 6851, 7398, ...}  A?????? 
24  {0, 1, 24, 69, 136, 225, 336, 469, 624, 801, 1000, 1221, 1464, 1729, 2016, 2325, 2656, 3009, 3384, 3781, 4200, 4641, 5104, 5589, 6096, 6625, 7176, 7749, ...}  A?????? 
25  {0, 1, 25, 72, 142, 235, 351, 490, 652, 837, 1045, 1276, 1530, 1807, 2107, 2430, 2776, 3145, 3537, 3952, 4390, 4851, 5335, 5842, 6372, 6925, 7501, 8100, ...}  A?????? 
26  {0, 1, 26, 75, 148, 245, 366, 511, 680, 873, 1090, 1331, 1596, 1885, 2198, 2535, 2896, 3281, 3690, 4123, 4580, 5061, 5566, 6095, 6648, 7225, 7826, 8451, ...}  A?????? 
27  {0, 1, 27, 78, 154, 255, 381, 532, 708, 909, 1135, 1386, 1662, 1963, 2289, 2640, 3016, 3417, 3843, 4294, 4770, 5271, 5797, 6348, 6924, 7525, 8151, 8802, ...}  A?????? 
28  {0, 1, 28, 81, 160, 265, 396, 553, 736, 945, 1180, 1441, 1728, 2041, 2380, 2745, 3136, 3553, 3996, 4465, 4960, 5481, 6028, 6601, 7200, 7825, 8476, 9153, ...}  A?????? 
29  {0, 1, 29, 84, 166, 275, 411, 574, 764, 981, 1225, 1496, 1794, 2119, 2471, 2850, 3256, 3689, 4149, 4636, 5150, 5691, 6259, 6854, 7476, 8125, 8801, 9504, ...}  A?????? 
30  {0, 1, 30, 87, 172, 285, 426, 595, 792, 1017, 1270, 1551, 1860, 2197, 2562, 2955, 3376, 3825, 4302, 4807, 5340, 5901, 6490, 7107, 7752, 8425, 9126, 9855, ...}  A?????? 
See also
Notes
 ↑ Author of the plots: Stefan Friedrich Birkner, License: Creative Commons AttributionShareAlike 3.0 Unported.
 ↑ Where is the ddimensional regular convex polytope number with 0dimensional facets, i.e. vertices V.
 ↑ Weisstein, Eric W., Polyhedral Formula, From MathWorldA Wolfram Web Resource.
 ↑ ^{4.0} ^{4.1} ^{4.2} Weisstein, Eric W., Lagrange's FourSquare Theorem, From MathWorldA Wolfram Web Resource.
 ↑ Weisstein, Eric W., Waring's Problem, From MathWorldA Wolfram Web Resource.
 ↑ Weisstein, Eric W., Fifteen Theorem, From MathWorldA Wolfram Web Resource.
 ↑ Where , k ≥ 1, n ≥ 0, is the ddimensional, d ≥ 0, (k+2)gonal base (hyper)pyramidal number where, for d ≥ 2, is the number of vertices (including the apex vertices) of the polygonal base (hyper)pyramid.
 ↑ Sondow, Jonathan and Weisstein, Eric W., Harmonic Number, From MathWorldA Wolfram Web Resource.
 ↑ Weisstein, Eric W., EulerMascheroni Constant, From MathWorldA Wolfram Web Resource.
 ↑ Weisstein, Eric W., Digamma Function, From MathWorldA Wolfram Web Resource.
 ↑ Weisstein, Eric W., Polygamma Function, From MathWorldA Wolfram Web Resource.
 ↑ Weisstein, Eric W., Relatively Prime, From MathWorldA Wolfram Web Resource.
 ↑ Weisstein, Eric W., Schläfli Symbol, From MathWorldA Wolfram Web Resource.
 ↑ Downey, Lawrence M., Ong, Boon W., and Sellers, James A., Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, 2008.
 ↑ PSYCHEDELIC GEOMETRY, INVERSE POLYGONAL NUMBERS SERIES
 ↑ GiSang Cheon and Moawwad E. A. ElMikkawy, GENERALIZED HARMONIC NUMBER IDENTITIES AND A RELATED MATRIX REPRESENTATION, J. Korean Math. Soc. 44 (2007), No. 2, pp. 487498.
 ↑ Sondow, Jonathan and Weisstein, Eric W., Riemann Zeta Function, From MathWorldA Wolfram Web Resource.
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.