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Orthoplex numbers

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The orthoplicial polytopic numbers are a family of sequences of figurate numbers corresponding to the
d
-dimensional regular cross-polytope (orthoplex) for each dimension
d
, where
d
is a nonnegative integer. These include the square gnomonic numbers, square numbers, the octahedral numbers and the hyperoctahedral numbers for
d > 3
. The cross-polytopes are duals of the orthotopes, the square being self dual (e.g. its dual is a diamond, which is a square rotated 45 degree).

The
d
-dimensional orthoplicial polytopic numbers, forming regular cross-polytopes (e.g. point, square gnomons, squares, octahedrons and then hyperoctahedrons)[1], where (−1)D-cells are the empty vertex set, 0D-cells are vertices, 1D-cells are edges, 2D-cells are faces, and so on...

d = 0
Regular 0-orthoplicial numbers Point numbers Form point (1 (-1)D-cell facets) (regular 0-orthoplex)
d = 1
Regular 1-orthoplicial numbers Henacross numbers (square gnomon numbers) Form henacross (square gnomons) (2 0D-cells facets) (regular 1-orthoplex)
d = 2
Regular 2-orthoplicial numbers Dicross numbers (square numbers) Form dicross (diamonds, i.e. squares) (4 1D-cells facets) (regular 2-orthoplex)
d = 3
Regular 3-orthoplicial numbers Tricross numbers (octahedral numbers) Form tricross (octahedrons) (8 2D-cells facets) (regular 3-orthoplex)
d = 4
Regular 4-orthoplicial numbers Tetracross numbers Form tetracross (hexadecachorons) (16 3D-cells facets) (regular 4-orthoplex)
d = 5
Regular 5-orthoplicial numbers Pentacross numbers Form pentacross (32-terons) (32 4D-cells facets) (regular 5-orthoplex)
d = 6
Regular 6-orthoplicial numbers Hexacross numbers Form hexacross (64-petons) (64 5D-cells facets) (regular 6-orthoplex)
d = 7
Regular 7-orthoplicial numbers Heptacross numbers Form heptacross (128-hexons) (128 6D-cells facets) (regular 7-orthoplex)
d = 8
Regular 8-orthoplicial numbers Octacross numbers Form octacross (256-heptons) (256 7D-cells facets) (regular 8-orthoplex)
...
... ... ... ... ...
d = d
Regular d-orthoplicial numbers (2d) (d-1)D-cells numbers Form d-cross (2d-(d-1)cells) (2d (d-1)-cells facets) (regular d-orthoplex)

All figurate numbers are accessible via this structured menu: Classifications of figurate numbers

Formulae

The th -dimensional orthoplicial polytopic number, with vertices and ()-dimensional facets, is given by the formula

[2]

where is the dimension and is the number of nondegenerate layered regular orthoplices ( giving no point and giving a single point, a degenerate regular orthoplex) of the -dimensional regular orthoplicial number (regular -orthoplex number.)

Formulae (from simplicial polytopic numbers)

The -dimensional orthoplicial polytopic numbers are obtained from the simplicial polytopic numbers by the formula

where is the th -dimensional regular simplicial polytopic number (regular simplex number.)

Formulae (from square hyperpyramidal numbers)

Since the square base of the square -dimensional (hyper)pyramid, for , is exposed in dimensions, we need to repeat the operation (of putting back to back the ()th resultant with the th resultant) times to obtain the orthoplicial polytopic numbers from the square hyperpyramidal numbers.

For , the th octahedral number is the th square dipyramidal number, i.e. the sum of the th and the ()th square pyramidal numbers, e.g.[3]

A000330
{0, 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, 1015, 1240, 1496, 1785, 2109, 2470, 2870, 3311, 3795, 4324, 4900, 5525, 6201, 6930, ...}
are the square 3D-hyperpyramidal numbers (square pyramidal numbers),
A005900
{0, 1, 6, 19, 44, 85, 146, 231, 344, 489, 670, 891, 1156, 1469, 1834, 2255, 2736, 3281, 3894, 4579, 5340, 6181, 7106, 8119, 9224, 10425, ...}
are the square 3D-hyperoctahedral numbers (octahedral numbers).
[4]

For , the th 4D-(semi)hyperoctahedral numbers are the sum of the th and the ()th square 4D-hyperpyramidal numbers, then the th 4D-hyperoctahedral numbers are the sum of the th and the ()th square 4D-(semi)hyperoctahedral numbers, e.g.

A002415
{0, 1, 6, 20, 50, 105, 196, 336, 540, 825, 1210, 1716, 2366, 3185, 4200, 5440, 6936, 8721, 10830, 13300, 16170, 19481, ...}
are the square 4D-hyperpyramidal numbers,
A006325
{0, 1, 7, 26, 70, 155, 301, 532, 876, 1365, 2035, 2926, 4082, 5551, 7385, 9640, 12376, 15657, 19551, 24130, 29470, ...}
are the square 4D-(semi)hyperoctahedral numbers,
A014820
{0, 1, 8, 33, 96, 225, 456, 833, 1408, 2241, 3400, 4961, 7008, 9633, 12936, 17025, 22016, 28033, 35208, 43681, 53600, ...}
are finally the square 4D-hyperoctahedral numbers.

For , the th 5D-(semisemi)hyperoctahedral numbers are the sum of the th and the ()th square 5D-hyperpyramidal numbers, the th 5D-(semi)hyperoctahedral numbers are the sum of the th and the ()th square 5D-(semisemi)hyperoctahedral numbers, then the th 5D-hyperoctahedral numbers are the sum of the th and the ()th square 5D-(semi)hyperoctahedral numbers, e.g.

A005585
{0, 1, 7, 27, 77, 182, 378, 714, 1254, 2079, 3289, 5005, 7371, 10556, 14756, 20196, 27132, 35853, 46683, 59983, 76153, ...}
are the square 5D-hyperpyramidal numbers,
A033455
{0, 1, 8, 34, 104, 259, 560, 1092, 1968, 3333, 5368, 8294, 12376, 17927, 25312, 34952, 47328, 62985, 82536, ...}
are the square 5D-(semisemi)hyperoctahedral numbers,
A061927
{0, 1, 9, 42, 138, 363, 819, 1652, 3060, 5301, 8701, 13662, 20670, 30303, 43239, 60264, 82280, 110313, 145521, ...}
are the square 5D-(semi)hyperoctahedral numbers,
A069038
{0, 1, 10, 51, 180, 501, 1182, 2471, 4712, 8361, 14002, 22363, 34332, 50973, 73542, 103503, 142544, 192593, 255834, ...}
are finally the square 5D-hyperoctahedral numbers.

For , the th 6D-(semi3)hyperoctahedral numbers are the sum of the th and the ()th square 6D-hyperpyramidal numbers, the th 6D-(semi2)hyperoctahedral numbers are the sum of the th and the ()th square 6D-(semi3)hyperoctahedral numbers, the th 6D-(semi)hyperoctahedral numbers are the sum of the th and the ()th square 6D-(semi2)hyperoctahedral numbers, then the th 6D-hyperoctahedral numbers are the sum of the th and the ()th square 6D-(semi)hyperoctahedral numbers, e.g.

A040977
{0, 1, 8, 35, 112, 294, 672, 1386, 2640, 4719, 8008, 13013, 20384, 30940, 45696, 65892, 93024, 128877, 175560, 235543, ...}
are the square 6D-hyperpyramidal numbers,
A??????
{0, 1, 9, 43, 147, 406, 966, 2058, 4026, 7359, 12727, ...}
are the square 6D-(semi3)hyperoctahedral numbers,
A??????
{0, 1, 10, 52, 190, 553, 1372, 3024, 6084, 11385, 20086, ...}
are the square 6D-(semi2)hyperoctahedral numbers,
A??????
{0, 1, 11, 62, 242, 743, 1925, 4396, 9108, 17469, 31471, ...}
are the square 6D-(semi)hyperoctahedral numbers,
A069039
{0, 1, 12, 73, 304, 985, 2668, 6321, 13504, 26577, 48940, 85305, 142000, 227305, 351820, 528865, 774912, 1110049, ...}
are finally the square 6D-hyperoctahedral numbers.

Thus, the -dimensional hyperoctahedral numbers expressed in terms of the -dimensional square hyperpyramidal numbers give

Formulae (cross-dimensional)

The th regular -dimensional orthoplicial polytopic numbers are given by the formulae[2][5]

Recurrence equation

with initial conditions

Method to obtain the recurrence equation from the generating function

By observing that

or

thus:

or

which reveals that

Generating function

Order of basis

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-polygonal numbers. Fermat claimed to have a proof of this result, although Fermat's proof has never been found.[6] Joseph Louis Lagrange proved the square case (known as the four squares theorem) 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-gon numbers (known as the polygonal number theorem,) while a vertical (higher dimensional), relative to the squares, generalization has also been made (known as the Hilbert-Waring problem.)[7]

A nonempty subset of nonnegative integers is called a basis of order if is the minimum number with the property that every nonnegative integer can be written as a sum of elements in . 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 k-gon numbers forms a basis of order , i.e. every nonnegative integer can be written as a sum of 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 such that every nonnegative integer is a sum of th powers, i.e. the set of th powers forms a basis of order . The Hilbert-Waring problem 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.

Pollock (1850) conjectured that every number is the sum of at most 5 octahedral numbers.[8]

Differences

Partial sums

Partial sums of reciprocals

Sum of reciprocals

Number of j-dimensional "vertices"

Table of formulae and values

are the number of vertices (0-dimensional), edges (1-dimensional), faces (2-dimensional), cells (3-dimensional)... respectively, where the ()-dimensional components are the actual facets. The regular orthoplicial numbers are listed by increasing number of vertices.

Regular orthoplicial numbers formulae and values
Name

Regular

-orthoplex

()-cell

Schläfli symbol[9]

Formulae


0 1 2 3 4 5 6 7 8 9 10 11 12 OEIS

number

0 Point

0-orthoplex

Hena-(-1)-cell

Nullcross

()

{}





0 1 1 1 1 1 1 1 1 1 1 1 1 A057427(),

for [10]

1 Square gnomon

1-orthoplex

Di-0-cell

Henacross

(2)

{}





0 1 2 3 4 5 6 7 8 9 10 11 12 A001477()
2 Square

2-orthoplex

Tetra-1-cell

Tetragon

Dicross

(4, 4)

{4}








0 1 4 9 16 25 36 49 64 81 100 121 144 A000290()
3 Octahedral

3-orthoplex

Octa-2-cell

Octahedron

Tricross

(6, 12, 8)

{3, 4}





0 1 6 19 44 85 146 231 344 489 670 891 1156 A005900()
4 Tetracross

4-orthoplex

24 3-cell

(8, 24, 32, 16)

{3, 3, 4}




0 1 8 33 96 225 456 833 1408 2241 3400 4961 7008 A014820()
5 Pentacross

5-orthoplex

25 4-cell

(10, 40, 80, 80, 32)

{3, 3, 3, 4}



0 1 10 51 180 501 1182 2471 4712 8361 14002 22363 34332 A069038()
6 Hexacross

6-orthoplex

26 5-cell

(12, 60, 160, 240, 192, 64)

{3, 3, 3, 3, 4}



0 1 12 73 304 985 2668 6321 13504 26577 48940 85305 142000 A069039()
7 Heptacross

7-orthoplex

27 6-cell

(14, 84, 280, 560, 672, 448, 128)

{3, 3, 3, 3, 3, 4}



0 1 14 99 476 1765 5418 14407 34232 74313 149830 284075 511380 A099193()
8 Octacross

8-orthoplex

28 7-cell

(16, 112, 448, 1120, 1792, 1792, 1024, 256)

{3, 3, 3, 3, 3, 3, 4}



0 1 16 129 704 2945 10128 29953 78592 187137 411280 845185 1640640 A099195()
9 Enneacross

9-orthoplex

29 8-cell

(18, 144, 672, 2016, 4032, 5376, 4608, 2304, 512)

{3, 3, 3, 3, 3, 3, 3, 4}



0 1 18 163 996 4645 17718 57799 166344 432073 1030490 2286955 4772780 A099196()
10 Decacross

10-orthoplex

210 9-cell

(20, 180, 960, 3360, 8064, 13440, 15360, 11520, 5120, 1024)

{3, 3, 3, 3, 3, 3, 3, 3, 4}



0 1 20 201 1360 7001 29364 104881 329024 927441 2390004 5707449 12767184 A099197()
11 Hendecacross

11-orthoplex

211 10-cell

(22, ..., 2048)

{3, 3, 3, 3, 3, 3, 3, 3, 3, 4}



0 1 22 243 1804 10165 46530 180775 614680 1871145 5188590 13286043 31760676 A??????
12 Dodecacross

12-orthoplex

212 11-cell

(24, ..., 4096)

{3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4}



0 1 24 289 2336 14305 71000 298305 1093760 3579585 10639320 29113953 74160672 A??????

Table of related formulae and values

are the number of vertices (0-dimensional), edges (1-dimensional), faces (2-dimensional), cells (3-dimensional)... respectively, where the ()-dimensional components are the actual facets. The regular orthoplicial numbers are listed by increasing number of vertices.

Regular orthoplicial numbers related formulae and values
d Name

Regular -orthoplex

()-cell

Schläfli symbol[9]

Generating

function



Order

of basis


[6][11][12]

Differences

Partial sums

Partial sums of reciprocals

Sum of reciprocals[13]

0 Point

0-orthoplex

Hena-(-1)-cell

()

{}





1 Square gnomon

1-orthoplex

di-0-cell

(2)

{}


[2]

 

2 Square

2-orthoplex

Tetra-1-cell

(4, 4)

{4}




Base 10: A013661
CFrac: A013679

3 Octahedral

3-orthoplex

Octa-2-cell

(6, 12, 8)

{3, 4}



Base 10: A175577

4 Tetracross

4-orthoplex

24 3-cell

(8, 24, 32, 16)

{3, 3, 4}


5 Pentacross

5-orthoplex

25 4-cell

(10, 40, 80, 80,
32)

{3, 3, 3, 4}


6 Hexacross

6-orthoplex

26 5-cell

(12, 60, 160, 240,
192, 64)

{3, 3, 3, 3, 4}


7 Heptacross

7-orthoplex

27 6-cell

(14, 84, 280, 560,
672, 448, 128)

{3, 3, 3, 3, 3, 4}


8 Octacross

8-orthoplex

28 7-cell

(16, 112, 448, 1120,
1792, 1792, 1024, 256)

{3, 3, 3, 3, 3, 3, 4}

9 Enneacross

9-orthoplex

29 8-cell

(18, ..., 512)

{3, 3, 3, 3, 3, 3,
3, 4}

10 Decacross

10-orthoplex

210 9-cell

(20, ..., 1024)

{3, 3, 3, 3, 3, 3,
3, 3, 4}

11 Hendecacross

11-orthoplex

211 10-cell

(22, ..., 2048)

{3, 3, 3, 3, 3, 3,
3, 3, 3, 4}

12 Dodecacross

12-orthoplex

212 11-cell

(24, ..., 4096)

{3, 3, 3, 3, 3, 3,
3, 3, 3, 3, 4}

Table of sequences

Orthoplicial polytopic numbers sequences
d
A-number
P  (d)2d(n), n   ≥   0
0 A057427
 (n), n   ≥   0
[10]
{0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...}
1 A001477
 (n)
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, ...}
2 A000290
 (n)
{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, 1089, 1156, 1225, 1296, ...}
3 A005900
 (n)
{0, 1, 6, 19, 44, 85, 146, 231, 344, 489, 670, 891, 1156, 1469, 1834, 2255, 2736, 3281, 3894, 4579, 5340, 6181, 7106, 8119, 9224, 10425, 11726, 13131, 14644, 16269, 18010, ...}
4 A014820
 (n  −  1)
{0, 1, 8, 33, 96, 225, 456, 833, 1408, 2241, 3400, 4961, 7008, 9633, 12936, 17025, 22016, 28033, 35208, 43681, 53600, 65121, 78408, 93633, 110976, 130625, 152776, 177633, ...}
5 A069038
 (n)
{0, 1, 10, 51, 180, 501, 1182, 2471, 4712, 8361, 14002, 22363, 34332, 50973, 73542, 103503, 142544, 192593, 255834, 334723, 432004, 550725, 694254, 866295, 1070904, ...}
6 A069039
 (n)
{0, 1, 12, 73, 304, 985, 2668, 6321, 13504, 26577, 48940, 85305, 142000, 227305, 351820, 528865, 774912, 1110049, 1558476, 2149033, 2915760, 3898489, 5143468, ...}
7 A099193
 (n)
{0, 1, 14, 99, 476, 1765, 5418, 14407, 34232, 74313, 149830, 284075, 511380, 880685, 1459810, 2340495, 3644272, 5529233, 8197758, 11905267, 16970060, ...}
8 A099195
 (n)
{0, 1, 16, 129, 704, 2945, 10128, 29953, 78592, 187137, 411280, 845185, 1640640, 3032705, 5373200, 9173505, 15158272, 24331777, 38058768, 58161793, 87037120, ...}
9 A099196
 (n)
{0, 1, 18, 163, 996, 4645, 17718, 57799, 166344, 432073, 1030490, 2286955, 4772780, 9446125, 17852030, 32398735, 56730512, 96220561, 158611106, 254831667, ...}
10 A099197
 (n)
{0, 1, 20, 201, 1360, 7001, 29364, 104881, 329024, 927441, 2390004, 5707449, 12767184, 26986089, 54284244, 104535009, 193664256, 346615329, 601446996, 1014889769, ...}
11 A??????
{0, 1, 22, 243, 1804, 10165, 46530, 180775, 614680, 1871145, 5188590, 13286043, 31760676, 71513949, 152784282, 311603535, 609802800, 1150082385, 2098144710, ...}
12 A??????
{0, 1, 24, 289, 2336, 14305, 71000, 298305, 1093760, 3579585, 10639320, 29113953, 74160672, 177435297, 401733528, 866121345, 1787527680, 3547412865, 6795639960, ...}

Orthoplicial polytopic numbers read cross-dimensionally

Note the disagreement about 0 0,[14] between the figurate number interpretation (which has to be 0 for
n = 0
) and the powers interpretation (which is 1).
Orthoplicial polytopic numbers read cross-dimensionally
b
A-number
P  (n)2n(b), b   ≥   0
0[14] A000004
 (n)
{0[15], 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...}
1 A000012
 (n)
{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...}
2 A004277
 (n)
{1, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, ...}
3 A058331
 (n)
{1, 3, 9, 19, 33, 51, 73, 99, 129, 163, 201, 243, 289, 339, 393, 451, 513, 579, 649, 723, 801, 883, 969, 1059, 1153, 1251, 1353, 1459, 1569, 1683, 1801, 1923, 2049, ...}
4 A??????
{1, 4, 16, 44, 96, 180, 304, 476, 704, 996, 1360, 1804, 2336, 2964, 3696, 4540, 5504, 6596, 7824, 9196, 10720, 12404, 14256, 16284, 18496, 20900, 23504, 26316, 29344, ...}
5 A??????
{1, 5, 25, 85, 225, 501, 985, 1765, 2945, 4645, 7001, 10165, 14305, 19605, 26265, 34501, 44545, 56645, 71065, 88085, 108001, 131125, 157785, 188325, 223105, 262501, ...}
6 A??????
{1, 6, 36, 146, 456, 1182, 2668, 5418, 10128, 17718, 29364, 46530, 71000, ...}
7 A??????
{1, 7, 49, 231, 833, 2471, 6321, 14407, 29953, 57799, 104881, 180775, 298305, ...}
8 A??????
{1, 8, 64, 344, 1408, 4712, 13504, 34232, 78592, 166344, 329024, 614680, 1093760, ...}
9 A??????
{1, 9, 81, 489, 2241, 8361, 26577, 74313, 187137, 432073, 927441, 1871145, 3579585, ...}
10 A??????
{1, 10, 100, 670, 3400, 14002, 48940, 149830, 411280, 1030490, 2390004, 5188590, 10639320, ...}
11 A??????
{1, 11, 121, 891, 4961, 22363, 85305, 284075, 845185, 2286955, 5707449, 13286043, 29113953, ...}
12 A??????
{1, 12, 144, 1156, 7008, 34332, 142000, 511380, 1640640, 4772780, 12767184, 31760676, 74160672, ...}
Orthoplicial polytopic numbers read cross-dimensionally (related formulae and values)
b
Formulae
Generating
function

Order
of basis
[6][11][16]
Differences

Partial sums
Partial sums of reciprocals
Sum of reciprocals[17]

0
1
2


3


4


5


6


7
8
9
10
11
12

See also

Notes

  1. Weisstein, Eric W., Hyperoctahedron, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/Hyperoctahedron.html]
  2. 2.0 2.1 2.2 Where is the -dimensional regular convex polytope number with vertices. Cite error: Invalid <ref> tag; name "d-dimensional_regular_convex_polytope_number_formula" defined multiple times with different content
  3. Weisstein, Eric W., Square Dipyramid, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/SquareDipyramid.html]
  4. Where , is the -dimensional, , ()-gonal base (hyper)pyramidal number where, for is the number of vertices (including the apex vertices) of the polygonal base (hyper)pyramid.
  5. Where is the -dimensional (hyper)-pyramidal number with vertices.
  6. 6.0 6.1 6.2 Weisstein, Eric W., Fermat's Polygonal Number Theorem, From MathWorld--A Wolfram Web Resource. Cite error: Invalid <ref> tag; name "FermatsPolygonalNumberTheorem" defined multiple times with different content
  7. Weisstein, Eric W., Waring's Problem, From MathWorld--A Wolfram Web Resource.
  8. Weisstein, Eric W., Pollock's Conjecture, From MathWorld--A Wolfram Web Resource.
  9. 9.0 9.1 Weisstein, Eric W., Schläfli Symbol, From MathWorld--A Wolfram Web Resource.
  10. 10.0 10.1 A057427 is the sign function (-1 for , 0 for , +1 for ,) while what we get here is the characteristic function of positive integers (0 for , +1 for .) Cite error: Invalid <ref> tag; name "chi_pos_ints" defined multiple times with different content
  11. 11.0 11.1 HYUN KWANG KIM, ON REGULAR POLYTOPE NUMBERS.
  12. Pollock, Frederick, On the extension of the principle of Fermat's theorem on the polygonal numbers to the higher order of series whose ultimate differences are constant. With a new theorem proposed, applicable to all the orders, Abstracts of the Papers Communicated to the Royal Society of London, 5 (1850) pp. 922-924.
  13. Downey, Lawrence M., Ong, Boon W., and Sellers, James A., Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, 2008.
  14. 14.0 14.1 Cf. 0 0 or the special case of zero to the zeroeth power.
  15. Note the disagreement about 0 0 between the figurate number interpretation (which has to be 0 for
    n = 0
    ) and the powers interpretation (which is 1).
  16. Pollock, Frederick, On the extension of the principle of Fermat’s theorem on the polygonal numbers to the higher order of series whose ultimate differences are constant. With a new theorem proposed, applicable to all the orders, Abstracts of the Papers Communicated to the Royal Society of London, 5 (1850) pp. 922–924.
  17. 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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