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Centered simplicial polytopic numbers
The centered simplicial polytopic numbers are a family of sequences of centered figurate numbers corresponding to the ddimensional simplex for each dimension d, where d is a nonnegative integer.
All figurate numbers are accessible via this structured menu: Classifications of figurate numbers
Contents
 1 Minimal nondegenerate polytopes in a ddimensional Euclidean space, d ≥ 0
 2 Formulae
 3 SchläfliPoincaré (convex) polytope formula
 4 Recurrence equation
 5 Generating function
 6 Order of basis
 7 Differences
 8 Partial sums
 9 Partial sums of reciprocals
 10 Sum of reciprocals
 11 Table of formulae and values
 12 Table of related formulae and values
 13 Table of sequences
 14 See also
 15 Notes
 16 External links
Minimal nondegenerate polytopes in a ddimensional Euclidean space, d ≥ 0
In a ddimensional Euclidean space in , d ≥ 0, the minimal number of vertices d + 1 gives the simplest dpolytope (the dsimplex,) i.e.:
 d = 0: the 0simplex (having 1 vertex) is the point (the 1 (1)cell, with 1 null polytope as facet)
 d = 1: the 1simplex (having 2 vertices) is the triangular gnomon (the 2 0cell, with 2 points as facets)
 d = 2: the 2simplex (having 3 vertices) is the trigon (triangle) (the 3 1cell, with 3 segments as facets)
 d = 3: the 3simplex (having 4 vertices) is the tetrahedron (the 4 2cell, with 4 faces as facets)
 d = 4: the 4simplex (having 5 vertices) is the pentachoron (the 5 3cell, with 5 rooms as facets)
 d = 5: the 5simplex (having 6 vertices) is the hexateron (the 6 4cell, with 6 4cells as facets)
 d = 6: the 6simplex (having 7 vertices) is the heptapeton (the 7 5cell, with 7 5cells as facets)
 d = 7: the 7simplex (having 8 vertices) is the octahexon (the 8 6cell, with 8 6cells as facets)
 d = 8: the 8simplex (having 9 vertices) is the enneahepton (the 9 7cell, with 9 7cells as facets)
 ...
 d = d: the dsimplex (having d+1 vertices) is the d+1 (d1)cell, with d+1 (d1)cells as facets
Formulae
The n^{th} ddimensional centered simplicial polytopic number is given by the formula:
 ^{[1]}
where d is the dimension.
SchläfliPoincaré (convex) polytope formula
Generalization for polytopes of DescartesEuler (convex) polyhedral formula:^{[2]}
where N_{0} is the number of 0dimensional elements, N_{1} is the number of 1dimensional elements, N_{2} is the number of 2dimensional elements...
Recurrence equation
with initial conditions
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 kpolygonal numbers. Fermat claimed to have a proof of this result, although Fermat's proof has never been found.^{[3]} 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 kgon numbers (known as the polygonal number theorem,) 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 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.
Differences
Partial sums
where is the m^{th} triangular number.
Partial sums of reciprocals
Sum of reciprocals
Table of formulae and values
N_{0}, N_{1}, N_{2},N_{3}, ... are the number of vertices (0dimensional), edges (1dimensional), faces (2dimensional), cells (3dimensional)... respectively, where the (n1)dimensional "vertices" are the actual facets. The centered simplicial polytopic numbers are listed by increasing number N_{0} of vertices.
d  Name
dsimplex d+1 (d1)cell (N_{0}, N_{1}, N_{2}, ...) Schläfli symbol^{[4]} 
Formulae

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

1  Centered 2step gnomonic
1simplex bi0cell (2) {} 

1  3  5  7  9  11  13  15  17  19  21  23  25  A005408 
2  Centered triangular
2simplex tri1cell (3, 3) {3} 
1  4  10  19  31  46  64  85  109  136  166  199  235  A005448  
3  Centered tetrahedral
3simplex tetra2cell (4, 6, 4) {3, 3} 
1  5  15  35  69  121  195  295  425  589  791  1035  1325  A005894  
4  Centered pentachoron
4simplex penta3cell (5, 10, 10, 5) {3, 3, 3} 
1  6  21  56  126  251  456  771  1231  1876  2751  3906  5396  A008498  
5  Centered hexateron
5simplex hexa4cell (6, 15, 20, 15, 6) {3, 3, 3, 3} 
1  7  28  84  210  462  923  1709  2975  4921  7798  11914  17640  A008499  
6  Centered heptapeton
6simplex hepta5cell (7, 21, 35, 35, 21, 7) {3, 3, 3, 3, 3} 
1  8  36  120  330  792  1716  3431  6427  11404  19328  31494  49596  A008500  
7  Centered octahexon
7simplex octa6cell (8, 28, 56, 70, 56, 28, 8) {3, 3, 3, 3, 3, 3} 
1  9  45  165  495  1287  3003  6435  12869  24301  43713  75417  125475  A008501  
8  Centered enneahepton
8simplex nona7cell (9, 36, 84, 126, 126, 84, 36, 9) {3, 3, 3, 3, 3, 3, 3} 
1  10  55  220  715  2002  5005  11440  24310  48619  92368  167905  293710  A008502  
9  Centered decaocton
9simplex deca8cell (10, 45, 120, 210, 252, 210, 120, 45, 10) {3, 3, 3, 3, 3, 3, 3, 3} 
1  11  66  286  1001  3003  8008  19448  43758  92378  184755  352705  646580  A008503  
10  Centered hendecaenneon
10simplex hendeca9cell (11, 55, 165, 330, 462, 462, 330, 165, 55, 11) {3, 3, 3, 3, 3, 3, 3, 3, 3} 
1  12  78  364  1365  4368  12376  31824  75582  167960  352716  705431  1352066  A008504  
11  Centered dodecadecon
11simplex dodeca10cell (12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12) {3, 3, 3, 3, 3, 3, 3, 3, 3, 3} 
1  13  91  455  1820  6188  18564  50388  125970  293930  646646  1352078  2704155  A008505  
12  Centered tridecahendecon
12simplex trideca11cell (13, ... Pascal's triangle 13^{th} row..., 13) {3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3} 
1  14  105  560  2380  8568  27132  77520  203490  497420  1144066  2496144  5200300  A008506 
N_{0}, N_{1}, N_{2},N_{3}, ... are the number of vertices (0dimensional), edges (1dimensional), faces (2dimensional), cells (3dimensional)... respectively, where the (n1)dimensional "vertices" are the actual facets. The centered simplicial polytopic numbers are listed by increasing number N_{0} of vertices.
d  Name
dsimplex d+1 (d1)cell (N_{0}, N_{1}, N_{2}, ...) Schläfli symbol^{[4]} 
Generating
function

Order
of basis ^{[3]}^{[5]}^{[6]} 
Differences

Partial sums

Partial sums of reciprocals

Sum of reciprocals^{[7]}


1  Centered 2step gnomonic
1simplex bi0cell (2) {} 

2  Centered triangular
2simplex tri1cell (3, 3) {3} 

3  Centered tetrahedral
3simplex tetra2cell (4, 6, 4) {3, 3} 

4  Centered pentachoron
4simplex penta3cell (5, 10, 10, 5) {3, 3, 3} 

5  Centered hexateron
5simplex hexa4cell (6, 15, 20, 15, 6) {3, 3, 3, 3} 

6  Centered heptapeton
6simplex hepta5cell (7, 21, 35, 35, 21, 7) {3, 3, 3, 3, 3} 

7  Centered octahexon
7simplex octa6cell (8, 28, 56, 70, 56, 28, 8) {3, 3, 3, 3, 3, 3} 

8  Centered enneahepton
8simplex nona7cell (9, 36, 84, 126, 126, 84, 36, 9) {3, 3, 3, 3, 3, 3, 3} 

9  Centered decaocton
9simplex deca8cell (10, 45, 120, 210, 252, 210, 120, 45, 10) {3, 3, 3, 3, 3, 3, 3, 3} 

10  Centered hendecaenneon
10simplex hendeca9cell (11, 55, 165, 330, 462, 462, 330, 165, 55, 11) {3, 3, 3, 3, 3, 3, 3, 3, 3} 

11  Centered dodecadecon
11simplex dodeca10cell (12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12) {3, 3, 3, 3, 3, 3, 3, 3, 3, 3} 

12  Centered tridecahendecon
12simplex trideca11cell (13, ... Pascal's triangle 13^{th} row..., 13) {3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3} 
Table of sequences
d  sequences 

2  {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, ...} 
3  {1, 5, 15, 35, 69, 121, 195, 295, 425, 589, 791, 1035, 1325, 1665, 2059, 2511, 3025, 3605, 4255, 4979, 5781, 6665, 7635, 8695, 9849, 11101, 12455, 13915, 15485, 17169, ...} 
4  {1, 6, 21, 56, 126, 251, 456, 771, 1231, 1876, 2751, 3906, 5396, 7281, 9626, 12501, 15981, 20146, 25081, 30876, 37626, 45431, 54396, 64631, 76251, 89376, 104131, ...} 
5  {1, 7, 28, 84, 210, 462, 923, 1709, 2975, 4921, 7798, 11914, 17640, 25416, 35757, 49259, 66605, 88571, 116032, 149968, 191470, 241746, 302127, 374073, 459179, 559181, ...} 
6  {1, 8, 36, 120, 330, 792, 1716, 3431, 6427, 11404, 19328, 31494, 49596, 75804, 112848, 164109, 233717, 326656, 448876, 607412, 810510, 1067760, 1390236, 1790643, ...} 
7  {1, 9, 45, 165, 495, 1287, 3003, 6435, 12869, 24301, 43713, 75417, 125475, 202203, 316767, 483879, 722601, 1057265, 1518517, 2144493, 2982135, 4088655, 5533155, ...} 
8  {1, 10, 55, 220, 715, 2002, 5005, 11440, 24310, 48619, 92368, 167905, 293710, 496705, 815188, 1302499, 2031535, 3100240, 4638205, 6814522, 9847045, 14013220, ...} 
9  {1, 11, 66, 286, 1001, 3003, 8008, 19448, 43758, 92378, 184755, 352705, 646580, 1143780, 1960255, 3265757, 5303727, 8416837, 13079352, 19937632, 29860259, 43999449, ...} 
10  {1, 12, 78, 364, 1365, 4368, 12376, 31824, 75582, 167960, 352716, 705431, 1352066, 2496066, 4457036, 7724795, 13033527, 21461804, 34565466, 54551718, 84504355, ...} 
11  {1, 13, 91, 455, 1820, 6188, 18564, 50388, 125970, 293930, 646646, 1352078, 2704155, 5200287, 9657609, 17383405, 30419935, 51889747, 86474661, 141070137, ...} 
12  {1, 14, 105, 560, 2380, 8568, 27132, 77520, 203490, 497420, 1144066, 2496144, 5200300, 10400599, 20058286, 37442055, 67863355, 119757470, 206244507, 347346468, ...} 
See also
Notes
 ↑ Where is the ddimensional centered regular convex polytope number with N_{0} vertices.
 ↑ Weisstein, Eric W., Polyhedral Formula, From MathWorldA Wolfram Web Resource.
 ↑ ^{3.0} ^{3.1} Weisstein, Eric W., Fermat's Polygonal Number Theorem, From MathWorldA Wolfram Web Resource.
 ↑ ^{4.0} ^{4.1} Weisstein, Eric W., Schläfli Symbol, From MathWorldA Wolfram Web Resource.
 ↑ HYUN KWANG KIM, ON REGULAR POLYTOPE NUMBERS.
 ↑ 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. 922924.
 ↑ 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, 2^{nd} ed., 1994.