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 kgonal number contains k more points than the previous layer.
Contents
Examples[]
Each sequence is a multiple of the triangular numbers plus 1. For example, the centered square numbers are four times the triangular numbers plus 1. In general, the centered kgonal numbers are k times the triangular numbers plus 1. Thus, the nth centered kgonal number is kn(n+1)/2+1
These series consist of the
 Centered triangular numbers 1, 4, X, 17, 27, 3X, 54, 71, 91, E4, 11X, 147, 177, 1XX, 224, 261, 2X1, 324, 36X, 3E7, 447, 49X, 534, 591, 631, ...
 Centered square numbers 1, 5, 11, 21, 35, 51, 71, 95, 101, 131, 165, 1X1, 221, 265, 2E1, 341, 395, 431, 491, 535, 5X1, 651, 705, 781, 841, ...
 Centered pentagonal numbers 1, 6, 14, 27, 43, 64, 8X, E9, 131, 16X, 1E0, 237, 287, 320, 37X, 421, 489, 53X, 5E4, 673, 737, 804, 896, 971, X51, ...
 Centered hexagonal numbers 1, 7, 17, 31, 51, 77, X7, 121, 161, 1X7, 237, 291, 331, 397, 447, 501, 581, 647, 717, 7E1, 891, 977, X67, E61, 1061, ... (which are also called hex numbers)
 Centered heptagonal numbers 1, 8, 1X, 37, 5E, 8X, 104, 145, 191, 224, 282, 327, 397, 452, 514, 5X1, 675, 754, 83X, 92E, X27, E2X, 1038, 1151, 1271, ...
 Centered octagonal numbers 1, 9, 21, 41, 69, X1, 121, 169, 201, 261, 309, 381, 441, 509, 5X1, 681, 769, 861, 961, X69, E81, 10X1, 1209, 1341, 1481, ... (which are exactly the odd squares)
 Centered nonagonal numbers 1, X, 24, 47, 77, E4, 13X, 191, 231, 29X, 354, 417, 4X7, 584, 66X, 761, 861, 96X, X84, EX7, 1117, 1254, 139X, 1531, 1691, ... (which include all even perfect numbers except 6)
 Centered dekragonal numbers 1, E, 27, 51, 85, 107, 157, 1E5, 261, 317, 39E, 471, 551, 63E, 737, 841, 955, X77, EX7, 1125, 1271, 1407, 156E, 1721, 18X1, ...
 Centered elpagonal numbers 1, 10, 2X, 57, 93, 11X, 174, 219, 291, 354, 426, 507, 5E7, 6E6, 804, 921, X49, E84, 110X, 1263, 1407, 157X, 1740, 1911, 1XE1, ...
 Centered dozagonal numbers 1, 11, 31, 61, X1, 131, 191, 241, 301, 391, 471, 561, 661, 771, 891, X01, E41, 1091, 1231, 13X1, 1561, 1731, 1911, 1E01, 2101, ... (which are also the star numbers)
and so on.
The star numbers are exactly the centered dozagonal numbers, thus, the star numbers are exactly the numbers obtained as the concatenation of a triangular number followed by a 1.
The triangular numbers are
 0, 1, 3, 6, X, 13, 19, 24, 30, 39, 47, 56, 66, 77, 89, X0, E4, 109, 123, 13X, 156, 173, 191, 1E0, 210, ...
and the star numbers are
 1, 11, 31, 61, X1, 131, 191, 241, 301, 391, 471, 561, 661, 771, 891, X01, E41, 1091, 1231, 13X1, 1561, 1731, 1911, 1E01, 2101, ...
Which are exactly the numbers obtained as the concatenation of a triangular number followed by a 1.
Thus, if we write a triangular number, and write a digit "1" after this number, then we get a star number.
Formula[]
As can be seen in the above diagrams, the nth centered kgonal number can be obtained by placing k copies of the (n−1)th triangular number around a central point; therefore, the nth centered kgonal number can be mathematically represented by
The difference of the nth and the (n+1)th consecutive centered kgonal numbers is k(2n+1).
Just as is the case with regular polygonal numbers, the first centered kgonal number is 1. Thus, for any k, 1 is both kgonal and centered kgonal. The next number to be both kgonal and centered kgonal can be found using the formula:
which tells us that X is both triangular and centered triangular, 21 is both square and centered square, etc.
Centered polygonal primes[]
Whereas a prime number p cannot be a polygonal number (except the trivial case, i.e. each p is the second pgonal number), many centered polygonal numbers are primes. In fact, if k ≥ 3, k ≠ 8, k ≠ 9, then there are infinitely many centered kgonal 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)
The centered polygonal primes are
 Centered triangular primes 17, 27, 91, 147, 2X1, 3E7, 447, 591, X41, 11X7, 1431, 1877, 2061, 2811, 2931, 40X7, 4531, 50E7, 5761, 6077, 6247, 7E01, 8717, 9371, 9591, X277, E6E7, E947, 11557, 13017, 13E31, 14X87, 15177, 16161, ...
 Centered square primes 5, 11, 35, 51, 95, 131, 221, 2E1, 431, 535, 705, 841, 905, 1011, 10E1, 1281, 1465, 1561, 1981, 2111, 2741, 2E51, 42X1, 4E71, 5711, 5905, 6101, 7395, 7X65, 8321, 8565, 8X41, 9591, E005, E561, E835, 101E1, 10781, 11X65, 12171, 13535, 15451, 17865, 19301, 19X71, 1X635, 20201, ...
 Centered pentagonal primes 27, 131, 237, 421, 737, 971, 1E17, 2X47, 3321, 34X1, 5367, 6957, 7491, 82E7, X107, X3E7, E0E1, 10467, 15EX1, 17777, 1EX81, 21497, 25961, 26261, ...
 Centered hexagonal primes (which are also called hex primes) 7, 17, 31, 51, X7, 1X7, 237, 291, 397, 447, 647, E61, 1061, 1167, 1391, 14E1, 1747, 1X01, 2097, 2537, 26X7, 2EX7, 3177, 3717, 4101, 4307, 4951, 5421, 5E37, 6947, 7001, 7817, 7XE1, 9777, E301, E647, 10EX7, 11721, 12277, 12651, 13217, 13X01, 173X7, 18531, 20961, 22591, 23417, 24801, 25147, 28421, 2X331, 2E307, ...
 Centered heptagonal primes 37, 5E, 145, 327, 397, 675, X27, 1151, 1647, 178E, 2181, 2847, 33X5, 4921, 5457, 62X5, 6571, 7207, 8515, X791, EX2E, 14E5E, 153E7, 18077, 19845, 1EX17, 2302E, 254E1, 2748E, 28E15, 29541, 318X5, 334EE, 35861, ...
 Centered octagonal primes (not exist, since 8n(n+1)/2+1 = (2n+1)^{2})
 Centered nonagonal primes (not exist, since 9n(n+1)/2+1 = (3n+1)×(3n+2))
 Centered dekragonal primes E, 27, 51, 85, 107, 157, 1E5, 471, 63E, 737, 841, 955, X77, 1125, 1407, 156E, 18X1, 1X6E, 2047, 2837, 3081, 3791, 3X31, 4357, 5585, 5891, 710E, 7E85, 8717, 8XE5, X711, XE3E, E377, E801, 10967, 11E87, 1246E, 12961, 1376E, 14087, 16355, 16901, 17277, 1783E, 1919E, 1X1X1, 1X7E5, 21367, 21X0E, 25987, 27721, 2899E, 29531, 2X83E, 2E3E7, 30755, 31E27, 32725, 33E47, 38171, 3E055, 3E941, 42051, 45161, 46857, 47607, 48385, 4X90E, ...
 Centered elpagonal primes 57, 291, 507, 5E7, 921, 1407, 1911, 1XE1, 24X7, 4E71, 6377, 7221, E327, 14891, 23297, 25781, 30947, 31577, 39551, 3X271, 47057, 4X531, 529E7, ...
 Centered dozagonal primes (which are also the star primes) 11, 31, 61, 131, 241, 301, 391, 471, 661, 771, 1231, 13X1, 1561, 1911, 1E01, 3291, 3541, 3801, 3X91, 4461, 4761, 5191, 5E91, 6331, 66X1, 7231, 7611, 7X01, 9261, X841, EX71, 10861, 12X91, 139X1, 16911, 17331, 18X31, 19491, 20961, 21E91, 23241, 26431, 27801, 29011, 2E0X1, 2E931, 31E01, 33471, 34161, 35771, 37E41, 3EE31, 42501, 44E61, 459X1, 47691, 4X291, 4E171, 50E61, 52991, 53901, 55791, 5E601, ...
and so on.
If Bunyakovsky conjecture is true, then centered kgonal primes exist (and there are infinitely many such primes) for all k except 8 and 9, and the numbers 8 and 9 are in the Catalan's conjecture (i.e. 8 and 9 are the only case of two consecutive perfect powers), besides, the product of 8 and 9 is 60, which is the smallest Achilles number, besides, the concatenation of 8 and 9 is 89, which is the smallest integer such that the factorization of over Q includes coefficients other than (i.e. the 89th cyclotomic polynomial, , is the first with coefficients other than ), besides, the 3smooth numbers (or the numbers n such that the reciprocal of n terminates) ≤ 10 are 1, 2, 3, 4, 6, 8, 9, and 10, all of these numbers except 8 and 9 are divisors of 10 (8 is because it has more prime factors 2 than 10, and 9 is because it has more prime factors 3 than 10) (thus, the numbers of digits of the reciprocal of all these n except 8 and 9 are all 1, while the numbers of digits of the reciprocal of 8 and 9 are 2).
Formulae[]
The n^{th} centered N_{0}gonal number, where n = 0 gives the central dot, is given by the formula:^{[1]}
where is the n^{th} triangular number.
SchläfliPoincaré (convex) polytope formula[]
SchläfliPoincaré generalization of the DescartesEuler (convex) polyhedral formula.^{[2]}
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 relation[]
with initial condition
Generating function[]
Differences[]
Partial sums[]
Partial sums of reciprocals[]
Sum of reciprocals[]
Table of formulae and values[]
N_{0}  Name  Formulae

n = 0  1  2  3  4  5  6  7  8  9  X  E  10  OEIS sequence 

3  Centered triangular 

1  4  X  17  27  3X  54  71  91  E4  11X  147  177  A005448 
4  Centered square 

1  5  11  21  35  51  71  95  101  131  165  1X1  221  A001844 
5  Centered pentagonal 

1  6  14  27  43  64  8X  E9  131  16X  1E0  237  287  A005891 
6  Centered hexagonal
Hex numbers 

1  7  17  31  51  77  X7  121  161  1X7  237  291  331  A003215 
7  Centered heptagonal 

1  8  1X  37  5E  8X  104  145  191  224  282  327  397  A069099 
8  Centered octagonal 
Odd squares 
1  9  21  41  69  X1  121  169  201  261  309  381  441  A016754 
9  Centered nonagonal 

1  X  24  47  77  E4  13X  191  231  29X  354  417  4X7  A060544 
X  Centered dekragonal 
Pentagram numbers 
1  E  27  51  85  107  157  1E5  261  317  39E  471  551  A062786 
E  Centered elpagonal 

1  10  2X  57  93  11X  174  219  291  354  426  507  5E7  A069125 
10  Centered dozagonal 
Star numbers Hexagram numbers 
1  11  31  61  X1  131  191  241  301  391  471  561  661  A003154 
11  Centered henadozagonal 

1  12  34  67  XE  144  1XX  265  331  40X  4E8  5E7  707  A069126 
12  Centered dodozagonal 
Heptagram numbers 
1  13  37  71  E9  157  207  289  361  447  543  651  771  A069127 
13  Centered tridozagonal 

1  14  3X  77  107  16X  224  2E1  391  484  58X  6X7  817  A069128 
14  Centered tetradozagonal 
Octagram numbers 
1  15  41  81  115  181  241  315  401  501  615  741  881  A069129 
15  Centered pentadozagonal 

1  16  44  87  123  194  25X  339  431  53X  660  797  927  A069130 
16  Centered hexadozagonal 
Enneagram numbers 
1  17  47  91  131  1X7  277  361  461  577  6X7  831  991  A069131 
17  Centered heptadozagonal 

1  18  4X  97  13E  1EX  294  385  491  5E4  732  887  X37  A069132 
18  Centered octadozagonal 
Dekragram numbers 
1  19  51  X1  149  211  2E1  3X9  501  631  779  921  XX1  A069133 
19  Centered enneadozagonal 

1  1X  54  X7  157  224  30X  411  531  66X  804  977  E47  A069178 
1X  Centered dekradozagonal 
Elpagram numbers 
1  1E  57  E1  165  237  327  435  561  6X7  84E  X11  EE1  A069173 
1E  Centered elpadozagonal 

1  20  5X  E7  173  24X  344  459  591  724  896  X67  1057  A069174 
20  Centered icosagonal 
Dozagram numbers 
1  21  61  101  181  261  361  481  601  761  921  E01  1101  A069190 
 ↑ Where is the ddimensional centered regular convex polytope number with N_{0} vertices.
 ↑ Weisstein, Eric W., Polyhedral Formula, From MathWorldA Wolfram Web Resource.