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Simplicial polytopic numbers
The simplicial polytopic numbers are a family of sequences of 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 Recurrence relation
 4 Generating function
 5 Simplicial polytopic numbers and Pascal's (rectangular) triangle columns
 6 Order of basis
 7 Differences
 8 Partial sums
 9 Partial sums of reciprocals
 10 Sum of reciprocals
 11 Number of jdimensional "vertices"
 12 Table of formulae and values
 13 Table of related formulae and values
 14 Table of sequences
 15 See also
 16 Notes
 17 External links
Minimal nondegenerate polytopes in a ddimensional Euclidean space, d ≥ 0
In a ddimensional Euclidean space , 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} simplicial dpolytopic numbers are given by the formulae ^{[1]} ^{[2]}^{[3]}:
 , or
where d ≥ 0 is the dimension and n1 ≥ 0 is the number of nondegenerate layered simplices (n1 = 0 giving a single dot, a degenerate simplex) of the ddimensional regular convex simplicial polytope number (dsimplex number.)
Recurrence relation
Generating function
Simplicial polytopic numbers and Pascal's (rectangular) triangle columns
n = 0  1  
1  1  1  
2  1  2  1  
3  1  3  3  1  
4  1  4  6  4  1  
5  1  5  10  10  5  1  
6  1  6  15  20  15  6  1  
7  1  7  21  35  35  21  7  1  
8  1  8  28  56  70  56  28  8  1  
9  1  9  36  84  126  126  84  36  9  1  
10  1  10  45  120  210  252  210  120  45  10  1  
11  1  11  55  165  330  462  462  330  165  55  11  1  
12  1  12  66  220  495  792  924  792  495  220  66  12  1 
d = 0  1  2  3  4  5  6  7  8  9  10  11  12 
d = 0  0simplicial numbers  Point numbers  Form point  (1 (1)cells "faces")  (0simplex) 
d = 1  1simplicial numbers  Linear numbers  Form segments  (2 0cells "faces")  (1simplex) 
d = 2  2simplicial numbers  Triangular numbers  Form triangles  (3 1cells "faces")  (2simplex) 
d = 3  3simplicial numbers  Tetrahedral numbers  Form tetrahedrons  (4 2cells "faces")  (3simplex) 
d = 4  4simplicial numbers  Pentachoron numbers  Form pentachorons  (5 3cells "faces")  (4simplex) 
d = 5  5simplicial numbers  Hexateron numbers  Form hexaterons  (6 4cells "faces")  (5simplex) 
d = 6  6simplicial numbers  Heptapeton numbers  Form heptapetons  (7 5cells "faces")  (6simplex) 
d = 7  7simplicial numbers  Octahexon numbers  Form octahexons  (8 6cells "faces")  (7simplex) 
d = 8  8simplicial numbers  Nonahepton numbers  Form nonaheptons  (9 7cells "faces")  (8simplex) 
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.^{[5]} 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 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 kgon numbers forms a basis of order , i.e. every nonnegative integer can be written as a sum of 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
Partial sums of reciprocals
Sum of reciprocals
The sum of reciprocals can be interpreted as , where is the probability that does not divide a random integer or the probability that two random integers and have different residues modulo .
The reciprocal of the sum of reciprocals thus gives:
Graphically, the second interpretation is the probability that a 2D integer lattice point does not lay on any straight line of the family , for integer values of . This family of functions consists of the bisection of the first and third quadrants and all its parallels at distances . When this includes all lattice points, hence .
Number of jdimensional "vertices"
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 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^{[6]} 
Formulae

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

0  Point
0simplex hena(1)cell () {} 
0  1  1  1  1  1  1  1  1  1  1  1  1  (NOT A057427) ^{[7]}  
1  Triangular gnomon
1simplex di0cell (2) {} 
^{[2]} 
0  1  2  3  4  5  6  7  8  9  10  11  12  A000027(n) 
2  Triangular
2simplex tri1cell (3, 3) {3} 
^{[2]} 
0  1  3  6  10  15  21  28  36  45  55  66  78  A000217(n) 
3  Tetrahedral
3simplex tetra2cell (4, 6, 4) {3, 3} 
^{[2]} 
0  1  4  10  20  35  56  84  120  165  220  286  364  A000292(n) 
4  Pentachoron
4simplex penta3cell (5, 10, 10, 5) {3, 3, 3} 
^{[2]} 
0  1  5  15  35  70  126  210  330  495  715  1001  1365  A000332(n+3) 
5  Hexateron
5simplex hexa4cell (6, 15, 20, 15, 6) {3, 3, 3, 3} 
^{[2]} 
0  1  6  21  56  126  252  462  792  1287  2002  3003  4368  A000389(n+4) 
6  Heptapeton
6simplex hepta5cell (7, 21, 35, 35, 21, 7) {3, 3, 3, 3, 3} 
^{[2]} 
0  1  7  28  84  210  462  924  1716  3003  5005  8008  12376  A000579(n+5) 
7  Octahexon
7simplex octa6cell (8, 28, 56, 70, 56, 28, 8) {3, 3, 3, 3, 3, 3} 
^{[2]} 
0  1  8  36  120  330  792  1716  3432  6435  11440  19448  31824  A000580(n+6) 
8  Enneahepton
8simplex nona7cell (9, 36, 84, 126, 126, 84, 36, 9) {3, 3, 3, 3, 3, 3, 3} 
^{[2]} 
0  1  9  45  165  495  1287  3003  6435  12870  24310  43758  75582  A000581(n+7) 
9  Decaocton
9simplex deca8cell (10, 45, 120, 210, 252, 210, 120, 45, 10) {3, 3, 3, 3, 3, 3, 3, 3} 
^{[2]} 
0  1  10  55  220  715  2002  5005  11440  24310  48620  92378  167960  A000582(n+8) 
10  Hendecaenneon
10simplex hendeca9cell (11, 55, 165, 330, 462, 462, 330, 165, 55, 11) {3, 3, 3, 3, 3, 3, 3, 3, 3} 
^{[2]} 
0  1  11  66  286  1001  3003  8008  19448  43758  92378  184756  352716  A001287(n+9) 
11  Dodecadecon
11simplex dodeca10cell (12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12) {3, 3, 3, 3, 3, 3, 3, 3, 3, 3} 
^{[2]} 
0  1  12  78  364  1365  4368  12376  31824  75582  167960  352716  705432  A001288(n+10) 
12  Tridecahendecon
12simplex trideca11cell (13, ... Pascal's triangle 13^{th} row..., 13) {3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3} 
^{[2]} 
0  1  13  91  455  1820  6188  18564  50388  125970  293930  646646  1352078  A010965(n+11) 
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 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^{[6]} 
Generating
function

Order
of basis

Differences

Partial sums

Partial sums of reciprocals

Sum of reciprocals^{[11]}


1  Triangular gnomon
1simplex bi0cell (2) {} 

^{[12]} [1]  
2  Triangular
2simplex tri1cell (3, 3) {3} 

[2] =


3  Tetrahedral
3simplex tetra2cell (4, 6, 4) {3, 3} 

[3] =

[4]  
4  Pentachoron
4simplex penta3cell (5, 10, 10, 5) {3, 3, 3} 

[5] =


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

[6]  
6  Heptapeton
6simplex hepta5cell (7, 21, 35, 35, 21, 7) {3, 3, 3, 3, 3} 

[7]  
7  Octahexon
7simplex octa6cell (8, 28, 56, 70, 56, 28, 8) {3, 3, 3, 3, 3, 3} 

[8]  
8  Enneahepton
8simplex nona7cell (9, 36, 84, 126, 126, 84, 36, 9) {3, 3, 3, 3, 3, 3, 3} 
[9]  
9  Decaocton
9simplex deca8cell (10, 45, 120, 210, 252, 210, 120, 45, 10) {3, 3, 3, 3, 3, 3, 3, 3} 
[10]  
10  Hendecaenneon
10simplex hendeca9cell (11, 55, 165, 330, 462, 462, 330, 165, 55, 11) {3, 3, 3, 3, 3, 3, 3, 3, 3} 
[11]  
11  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]  
12  Tridecahendecon
12simplex trideca11cell (13, ... Pascal's triangle 13^{th} row..., 13) {3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3} 
[13] 
Table of sequences
d  sequences 

1  {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, 48, ...} 
2  {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, 630, 666, 703, 741, ...} 
3  {0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, 1140, 1330, 1540, 1771, 2024, 2300, 2600, 2925, 3276, 3654, 4060, 4495, 4960, 5456, 5984, ...} 
4  {0, 1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, 1365, 1820, 2380, 3060, 3876, 4845, 5985, 7315, 8855, 10626, 12650, 14950, 17550, 20475, 23751, 27405, 31465, 35960, ...} 
5  {0, 1, 6, 21, 56, 126, 252, 462, 792, 1287, 2002, 3003, 4368, 6188, 8568, 11628, 15504, 20349, 26334, 33649, 42504, 53130, 65780, 80730, 98280, 118755, 142506, 169911, ...} 
6  {0, 1, 7, 28, 84, 210, 462, 924, 1716, 3003, 5005, 8008, 12376, 18564, 27132, 38760, 54264, 74613, 100947, 134596, 177100, 230230, 296010, 376740, 475020, 593775, ...} 
7  {0, 1, 8, 36, 120, 330, 792, 1716, 3432, 6435, 11440, 19448, 31824, 50388, 77520, 116280, 170544, 245157, 346104, 480700, 657800, 888030, 1184040, 1560780, 2035800, ...} 
8  {0, 1, 9, 45, 165, 495, 1287, 3003, 6435, 12870, 24310, 43758, 75582, 125970, 203490, 319770, 490314, 735471, 1081575, 1562275, 2220075, 3108105, 4292145, 5852925, ...} 
9  {0, 1, 10, 55, 220, 715, 2002, 5005, 11440, 24310, 48620, 92378, 167960, 293930, 497420, 817190, 1307504, 2042975, 3124550, 4686825, 6906900, 10015005, 14307150, ...} 
10  {0, 1, 11, 66, 286, 1001, 3003, 8008, 19448, 43758, 92378, 184756, 352716, 646646, 1144066, 1961256, 3268760, 5311735, 8436285, 13123110, 20030010, 30045015, ... } 
11  {0, 1, 12, 78, 364, 1365, 4368, 12376, 31824, 75582, 167960, 352716, 705432, 1352078, 2496144, 4457400, 7726160, 13037895, 21474180, 34597290, 54627300, ...} 
12  {0, 1, 13, 91, 455, 1820, 6188, 18564, 50388, 125970, 293930, 646646, 1352078, 2704156, 5200300, 9657700, 17383860, 30421755, 51895935, 86493225, 141120525, ...} 
See also
Centered simplicial polytopic numbers
Notes
 ↑ Where is the ddimensional regular convex polytope number with N_{0} 0dimensional elements (vertices V.)
 ↑ ^{2.00} ^{2.01} ^{2.02} ^{2.03} ^{2.04} ^{2.05} ^{2.06} ^{2.07} ^{2.08} ^{2.09} ^{2.10} ^{2.11} ^{2.12} Weisstein, Eric W., Rising Factorial, From MathWorldA Wolfram Web Resource.
 ↑ Weisstein, Eric W., Multichoose, From MathWorldA Wolfram Web Resource.
 ↑ Weisstein, Eric W., Simplex, From MathWorldA Wolfram Web Resource.
 ↑ Weisstein, Eric W., Fermat's Polygonal Number Theorem, From MathWorldA Wolfram Web Resource.
 ↑ ^{6.0} ^{6.1} Weisstein, Eric W., Schläfli Symbol, From MathWorldA Wolfram Web Resource.
 ↑ A057427 is the sign function (1 for n < 0, 0 for n = 0, +1 for n > 0,) while what we get here is the characteristic function of positive integers (0 for n ≤ 0, +1 for n ≥ 1.)
 ↑ Weisstein, Eric W., Fermat's Polygonal Number Theorem, 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.
 ↑ Sondow, Jonathan and Weisstein, Eric W., Harmonic Number, 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, 2^{nd} ed., 1994.