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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 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 k-gonal numbers are k times the triangular numbers plus 1. Thus, the nth centered k-gonal 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 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

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

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:

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 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)

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 k-gonal 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 3-smooth 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 nth centered N0-gonal number, where n = 0 gives the central dot, is given by the formula:[1]

where is the nth triangular number.

Schläfli-Poincaré (convex) polytope formula[]

Schläfli-Poincaré generalization of the Descartes-Euler (convex) polyhedral formula.[2]

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[]

Differences[]

Partial sums[]

Partial sums of reciprocals[]

Sum of reciprocals[]

Table of formulae and values[]

Centered polygonal numbers formulae and values
N0 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
  1. Where is the d-dimensional centered regular convex polytope number with N0 vertices.
  2. Weisstein, Eric W., Polyhedral Formula, From MathWorld--A Wolfram Web Resource.