There are six regular convex polychora (4-dimensional hyper-solids,) which, except for the 24-cell, are the analogues of the Platonic solids. Listed by increasing number of vertices, the six regular convex polychora are:
- the 5 vertices (self dual) 5-cell (pentachoron) or 4-simplex (a hyper-tetrahedron,)
- the 8 vertices 16-cell or 4-cross polytope or 4-orthoplex (a hyper-octahedron,)
- the 16 vertices 8-cell or 4-cube or 4-orthotope or tesseract (a hyper-cube,)
- the 24 vertices (self dual) 24-cell, which has no perfect analogy in higher or lower dimensional spaces,
- the 120 vertices 600-cell (a hyper-icosahedron,)
- and the 600 vertices 120-cell (a hyper-dodecahedron.)
The 5-cell and 24-cell are self-dual, the 16-cell is the dual of the 8-cell, and the 600- and 120-cells are dual to each other.
The number of cells, faces, edges and vertices, for each of the six regular convex polychora are give the sequences:
- A063924 Number of cells (3-dimensional elements) in the regular 4-dimensional polytopes.
- A063925 Number of faces (2-dimensional elements) in the regular 4-dimensional polytopes.
- A063926 Number of edges (1-dimensional elements) in the six regular 4-dimensional polytopes.
- A063927 Number of vertices (0-dimensional elements) in the regular 4-dimensional polytopes.
The polychoron numbers are the numbers of dots in a layered geometric arrangement into one of the 6 regular convex polychoron hyper-solids.[1] The polychoron numbers start with one initial dot (n = 1,) then with one dot at each vertex of a given polychoron hyper-solid (n = 2,) with each of the following layers growing out of the initial vertex with one more dot per edge than the preceding layer, and where overlapping dots are counted only once.
The 6 types of regular convex polychoron numbers are:
- A000332(n+3) Pentachoron numbers, binomial(n+3,4).
- A000583 The tesseract numbers, fourth powers: n^4.
- A092182 Figurate numbers based on the 600-cell (4-D polytope with Schlaefli symbol {3,3,5}).
- A092183 Figurate numbers based on the 120-cell (4-D polytope with Schlaefli symbol {5,3,3}).
- A092181 Figurate numbers based on the 24-cell (4-D polytope with Schlaefli symbol {3,4,3}).
All figurate numbers are accessible via this structured menu: Classifications of figurate numbers
Formulae
The nth 4-dimensional N3-cell regular polytope (having N0 vertices) number is given by the formula:[2]
![{\displaystyle P_{N_{0}}^{(4)}(n)={\binom {n+3}{4}}+[N_{0}-5]{\binom {n+2}{4}}+[???]{\binom {n+1}{4}}+[???]{\binom {n}{4}},\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/ca2d3ab46a5800f9397a0b695a9987a9996acd48)
For the 5-cell numbers:
![{\displaystyle P_{5}^{(4)}(n)={\binom {n+3}{4}}+[N_{0}-5]{\binom {n+2}{4}}+[N_{0}-5]{\binom {n+1}{4}}+[0]{\binom {n}{4}},\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/5092f112ed23566e1a2692bae2cc9afb97c39957)
For the 16-cell and 8-cell numbers:
![{\displaystyle P_{8}^{(4)}(n)={\binom {n+3}{4}}+[N_{0}-5]{\binom {n+2}{4}}+[N_{0}-5]{\binom {n+1}{4}}+[1]{\binom {n}{4}},\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/f44610c97d96ce1741f8894007a201967f5c674c)
For the 24-cell numbers:
![{\displaystyle P_{24}^{(4)}(n)={\binom {n+3}{4}}+[N_{0}-5]{\binom {n+2}{4}}+[43]{\binom {n+1}{4}}+[9]{\binom {n}{4}},\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/a16d52977ed10f5198f02c4756a86adb0c08e208)
For the 600-cell and 120-cell numbers:
![{\displaystyle P_{120}^{(4)}(n)={\binom {n+3}{4}}+[N_{0}-5]{\binom {n+2}{4}}+[357]{\binom {n+1}{4}}+[107]{\binom {n}{4}},\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/da8d1758fa16f0060687cdc6f410c95649aabf7b)
Schläfli-Poincaré (convex) polytope formula
Schläfli-Poincaré generalization of the Descartes-Euler (convex) polyhedral formula.[3]
For 4-dimensional (d = 4) regular convex polytopes:
![{\displaystyle {\sum _{i=0}^{d-1}(-1)^{i}N_{i}}=N_{0}-N_{1}+N_{2}-N_{3}=V-E+F-C=0,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/c635596a4701482919c45174e062b62c20597aad)
where N0 is the number of 0-dimensional elements (vertices V), N1 is the number of 1-dimensional elements (edges V), N2 is the number of 2-dimensional elements (faces F) and N3 is the number of 3-dimensional elements (cells C) of the regular convex polytope.
Recurrence equation
The recurrence equation seems to be the same for all polychoron numbers (TO BE VERIFIED for 5-cell, 16-cell and 8-cell numbers):
![{\displaystyle P_{N_{0}}^{(4)}(n)=5P_{N_{0}}^{(4)}(n-1)-10P_{N_{0}}^{(4)}(n-2)+10P_{N_{0}}^{(4)}(n-3)-5P_{N_{0}}^{(4)}(n-4)+P_{N_{0}}^{(4)}(n-5)\,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/730c5ff86ef057858a71c3e7f70516da7ce73eb8)
with initial conditions
![{\displaystyle P_{N_{0}}^{(4)}(0)=?\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/e782bbe87b23622ef715b356f5fa53cc580b9106)
For the 5-cell numbers:
![{\displaystyle P_{5}^{(4)}(n)=,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/bae43f4602e5bbf1dc6d8c9ac8031f642df2720a)
For the 16-cell and 8-cell numbers:
![{\displaystyle P_{8}^{(4)}(n)=,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/c04b9019ed4585a56f8a6ba183297daf4cf0c0f9)
![{\displaystyle P_{16}^{(4)}(n)=,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/725c1295b0e9f3f9a8ff8a72ed47693f36f6c4fa)
For the 24-cell numbers:
![{\displaystyle P_{24}^{(4)}(n)=5P_{24}^{(4)}(n-1)-10P_{24}^{(4)}(n-2)+10P_{24}^{(4)}(n-3)-5P_{24}^{(4)}(n-4)+P_{24}^{(4)}(n-5),\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/f95a3505b8816cd601735af8d90a5c4249101a79)
For the 600-cell and 120-cell numbers:
![{\displaystyle P_{120}^{(4)}(n)=5P_{120}^{(4)}(n-1)-10P_{120}^{(4)}(n-2)+10P_{120}^{(4)}(n-3)-5P_{120}^{(4)}(n-4)+P_{120}^{(4)}(n-5),\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/da36d83302d5f43dc8cc2ed31f64557986084ce2)
![{\displaystyle P_{600}^{(4)}(n)=5P_{600}^{(4)}(n-1)-10P_{600}^{(4)}(n-2)+10P_{600}^{(4)}(n-3)-5P_{600}^{(4)}(n-4)+P_{600}^{(4)}(n-5).\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/fcbfb6666307dd0a0bbc057db09d58eb279c3279)
Ordinary generating function
![{\displaystyle G_{\{P_{N_{0}}^{(4)}(n)\}}(x)={\frac {x(1+[N_{0}-5]x+[???]x^{2}+[???]x^{3})}{(1-x)^{5}}}\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/c4a64baeddec1620078266ee3f697246cc2e6833)
For the 5-cell numbers:
![{\displaystyle G_{\{P_{5}^{(4)}(n)\}}(x)={\frac {x}{(1-x)^{5}}}={\frac {x(1+[N_{0}-5]x+[N_{0}-5]x^{2}+0x^{3})}{(1-x)^{5}}},\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/9d8afe8e92708ece806c5aae4801d6caa941fa35)
For the 16-cell and 8-cell numbers:
![{\displaystyle G_{\{P_{8}^{(4)}(n)\}}(x)={\frac {x(1+3x+3x^{2}+x^{3})}{(1-x)^{5}}}={\frac {x(1+[N_{0}-5]x+[N_{0}-5]x^{2}+x^{3})}{(1-x)^{5}}},\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/31741a3ba763b0b7d77199b368490deecb94b50f)
![{\displaystyle G_{\{P_{16}^{(4)}(n)\}}(x)={\frac {x(1+11x+11x^{2}+x^{3})}{(1-x)^{5}}}={\frac {x(1+[N_{0}-5]x+[N_{0}-5]x^{2}+x^{3})}{(1-x)^{5}}},\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/4d9f02d51a7f3b990a2ad327b48b4024f998e5b4)
For the 24-cell numbers:
![{\displaystyle G_{\{P_{24}^{(4)}(n)\}}(x)={\frac {x(9x^{3}+43x^{2}+19x+1)}{(1-x)^{5}}}={\frac {x(1+[N_{0}-5]x+43x^{2}+9x^{3})}{(1-x)^{5}}},\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/b731712ea3b56f3bcaab051d5d22626321144e0c)
For the 600-cell and 120-cell numbers:
![{\displaystyle G_{\{P_{120}^{(4)}(n)\}}(x)={\frac {x(107x^{3}+357x^{2}+115x+1)}{(1-x)^{5}}}={\frac {x(1+[N_{0}-5]x+357x^{2}+107x^{3})}{(1-x)^{5}}},\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/def59b317c711012e93c0f3aaebd538682c41f96)
![{\displaystyle G_{\{P_{600}^{(4)}(n)\}}(x)={\frac {x(543x^{3}+1993x^{2}+595x+1)}{(1-x)^{5}}}={\frac {x(1+[N_{0}-5]x+1993x^{2}+543x^{3})}{(1-x)^{5}}},\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/304774c40a8159eb344a320a91c184c2207f1c22)
Procedure to obtain the generating functions
![{\displaystyle G_{\{P_{24}^{(4)}(n)\}}(x)=G_{\{n^{2}(3n^{2}-4n+2)\}}(x)=G_{\{3n^{4}-4n^{3}+2n^{2}\}}(x)=3G_{\{n^{4}\}}(x)-4G_{\{n^{3}\}}(x)+2G_{\{n^{2}\}}(x)\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/382ef7abc93488f7261425802566ba13493a7a2c)
![{\displaystyle ={\frac {x}{(1-x)^{5}}}\{3(1+11x+11x^{2}+x^{3})-4(1-x)(1+4x+x^{2})+2(1-x)^{2}(1+x)\}={\frac {x(9x^{3}+43x^{2}+19x+1)}{(1-x)^{5}}},\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/04cd5506a71ebdcec58a3cb7e3e90052c175a665)
![{\displaystyle G_{\{P_{120}^{(4)}(n)\}}(x)=G_{\{{\frac {n(145n^{3}-280n^{2}+179n-38)}{6}}\}}(x)={\frac {1}{6}}G_{\{145n^{4}-280n^{3}+179n^{2}-38n\}}(x)={\frac {1}{6}}\{145G_{\{n^{4}\}}(x)-280G_{\{n^{3}\}}(x)+179G_{\{n^{2}\}}(x)-38G_{\{n\}}(x)\}\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/83f2ce9fa082275b5ab71cc221913696da614887)
![{\displaystyle ={\frac {x}{6(1-x)^{5}}}\{145(1+11x+11x^{2}+x^{3})-280(1-x)(1+4x+x^{2})+179(1-x)^{2}(1+x)-38(1-x)^{3}\}={\frac {x(107x^{3}+357x^{2}+115x+1)}{(1-x)^{5}}},\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/d017c2b0fc5a3d2c6f7c5ca2f7104fcfc797c1c1)
![{\displaystyle G_{\{P_{600}^{(4)}(n)\}}(x)=G_{\{{\frac {n(261n^{3}-504n^{2}+283n-38)}{2}}\}}(x)={\frac {1}{2}}G_{\{261n^{4}-504n^{3}+283n^{2}-38n\}}(x)={\frac {1}{2}}\{261G_{\{n^{4}\}}(x)-504G_{\{n^{3}\}}(x)+283G_{\{n^{2}\}}(x)-38G_{\{n\}}(x)\}\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/97ada6cd79c36c7933e4e9b4294332e38b3ffa88)
![{\displaystyle ={\frac {x}{2(1-x)^{5}}}\{261(1+11x+11x^{2}+x^{3})-504(1-x)(1+4x+x^{2})+283(1-x)^{2}(1+x)-38(1-x)^{3}\}={\frac {x(543x^{3}+1993x^{2}+595x+1)}{(1-x)^{5}}},\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/691e6e13d06ba88de18b7ddc8600a83259948cbf)
where (Cf. Generating function for regular orthotopic numbers)
![{\displaystyle G_{\{n^{4}\}}(x)={\frac {x\ E_{4}(x)}{(1-x)^{5}}}={\frac {x\ (1+11x+11x^{2}+x^{3})}{(1-x)^{5}}},\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/3f7dab1153271f5b0487a92eaad084394a3a787c)
![{\displaystyle G_{\{n^{3}\}}(x)={\frac {x\ E_{3}(x)}{(1-x)^{4}}}={\frac {x\ (1+4x+x^{2})}{(1-x)^{4}}},\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/7426cb990004bf1453abbc76306d3f5450ff184d)
![{\displaystyle G_{\{n^{2}\}}(x)={\frac {x\ E_{2}(x)}{(1-x)^{3}}}={\frac {x\ (1+x)}{(1-x)^{3}}},\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/9d224d2ac794f3651d6d1cfcbbf24a9d444873b2)
![{\displaystyle G_{\{n\}}(x)={\frac {x\ E_{1}(x)}{(1-x)^{2}}}={\frac {x\ (1)}{(1-x)^{2}}}.\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/17cb0d88c801c7d63dfaeacae1115b9b500e19a8)
Exponential generating function
![{\displaystyle E_{\{P_{N_{0}}^{(4)}(n)\}}(x)=?\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/9df97ac7849f85d38d1dce644c34b889e4b5919c)
For the 5-cell numbers:
![{\displaystyle E_{\{P_{5}^{(4)}(n)\}}(x)=,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/1c3e9353ba3848fa578433e5c7f286a9570a3379)
For the 16-cell and 8-cell numbers:
![{\displaystyle E_{\{P_{8}^{(4)}(n)\}}(x)=,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/636de0fd6d4779488755eace372d8ea9ce337b06)
For the 24-cell numbers:
![{\displaystyle E_{\{P_{24}^{(4)}(n)\}}(x)=,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/27d51511161a34a121eca9df3f79a49410de0246)
For the 600-cell and 120-cell numbers:
![{\displaystyle E_{\{P_{120}^{(4)}(n)\}}(x)=,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/935918836f6551c656cd85dc2f3418f57ef1f9da)
![{\displaystyle E_{\{P_{600}^{(4)}(n)\}}(x)=,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/abcc590698299ccc86da82a97d7908bf8a9d18bd)
Dirichlet generating function
![{\displaystyle D_{\{P_{N_{0}}^{(4)}(n)\}}(x)=?\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/ab2ff61ff334bae92ec69f490802cfdce3316de8)
For the 5-cell numbers:
![{\displaystyle D_{\{P_{5}^{(4)}(n)\}}(x)=,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/5b7f21c82429ffd1101fc26c5b3615621891c045)
For the 16-cell and 8-cell numbers:
![{\displaystyle D_{\{P_{8}^{(4)}(n)\}}(x)=,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/243ef67bc4afec1ee67232061a8f709d516eda2f)
[4]
For the 24-cell numbers:
![{\displaystyle D_{\{P_{24}^{(4)}(n)\}}(x)=,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/a1cff03f4c0fa457a4f8767a045b9378dbf83458)
For the 600-cell and 120-cell numbers:
![{\displaystyle D_{\{P_{120}^{(4)}(n)\}}(x)=,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/9324e96a8b4d96581fcf1ad0cd24c94958fc3f14)
![{\displaystyle D_{\{P_{600}^{(4)}(n)\}}(x)=,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/bd302e2bdceb5a34a4900ac452c863059a95776a)
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.[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 k-gon numbers (known as the polygonal number theorem,) while a vertical (higher dimensional) generalization has also been made (known as the Hilbert-Waring 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 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.
Differences
Partial sums
The partial sums correspond to 5-dimensional polychoron hyperpyramidal numbers.
![{\displaystyle \sum _{n=1}^{m}P_{N_{0}}^{(4)}(n)=?,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/2db4ed83a4b19a05f55045d20a2f89908affdbe7)
Partial sums of reciprocals
Sum of reciprocals
Table of formulae and values
N0, N1, N2 and N3 are the number of vertices (0-dimensional elements), edges (1-dimensional elements), faces (2-dimensional elements) and cells (3-dimensional elements) respectively, where the cells are the actual facets. The regular polychorons are listed by increasing number N0 of vertices.
Regular polychoron numbers formulae and values
Rank
r
|
N0
|
Name
(N0, N1, N2, N3)
Schläfli symbol[6]
|
Formulae
|
Generating
function
|
n = 0
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
12
|
OEIS
number
|
0
|
5
|
Pentachoron
5 cell
(5, 10, 10, 5)
{3, 3, 3}
|
[7]
|
|
0
|
1
|
5
|
15
|
35
|
70
|
126
|
210
|
330
|
495
|
715
|
1001
|
1365
|
A000332(n+3)
|
1
|
8
|
16 cell
(8, 24, 32, 16)
{3, 3, 4}
|
|
|
0
|
1
|
8
|
33
|
96
|
225
|
456
|
833
|
1408
|
2241
|
3400
|
4961
|
7008
|
A014820(n-1)
|
2
|
16
|
Tesseract
8 cell
(16, 32, 24, 8)
{4, 3, 3}
|
|
|
0
|
1
|
16
|
81
|
256
|
625
|
1296
|
2401
|
4096
|
6561
|
10000
|
14641
|
20736
|
A000583
|
3
|
24
|
24 cell
(24, 96, 96, 24)
{3, 4, 3}
|
|
|
0
|
1
|
24
|
153
|
544
|
1425
|
3096
|
5929
|
10368
|
16929
|
26200
|
38841
|
55584
|
A092181
|
4
|
120
|
600 cell
(120, 720, 1200, 600)
{3, 3, 5}
|
|
|
0
|
1
|
120
|
947
|
3652
|
9985
|
22276
|
43435
|
76952
|
126897
|
197920
|
295251
|
424700
|
A092182
|
5
|
600
|
120 cell
(600, 1200, 720, 120)
{5, 3, 3}
|
|
|
0
|
1
|
600
|
4983
|
19468
|
53505
|
119676
|
233695
|
414408
|
683793
|
1066960
|
1592151
|
2290740
|
A092183
|
Table of related formulae and values
N0, N1, N2 and N3 are the number of vertices (0-dimensional elements), edges (1-dimensional elements), faces (2-dimensional elements) and cells (3-dimensional elements) respectively, where the cells are the actual facets. The regular polychorons are listed by increasing number N0 of vertices.
Regular polychoron numbers related formulae and values
Rank
r
|
N0
|
Name
(N0, N1, N2, N3)
Schläfli symbol[6]
|
Order
of basis
|
Differences
|
Partial sums
|
Partial sums of reciprocals
|
Sum of Reciprocals[8][9]
|
0
|
5
|
Pentachoron
5 cell
(5, 10, 10, 5)
{3, 3, 3}
|
[10]
|
Tetrahedral numbers
(A000292(n))
|
[7]
[11]
A000389
|
|
|
1
|
8
|
16 cell
(8, 24, 32, 16)
{3, 3, 4}
|
[10]
|
Centered octahedral numbers
(A001845(n-1))
|
A061927(n-1)?
|
|
|
2
|
16
|
Tesseract
8 cell
(16, 32, 24, 8)
{4, 3, 3}
|
|
[12]
Centered cubic(n) +
6 Square pyramidal(n-1)
Rhombic dodecahedral numbers [13]
(A005917(n-1))
|
A000538
|
|
[4]
|
3
|
24
|
24 cell
(24, 96, 96, 24)
{3, 4, 3}
|
[10]
|
|
|
|
|
4
|
120
|
600 cell
(120, 720, 1200, 600)
{3, 3, 5}
|
[10]
|
|
|
|
|
5
|
600
|
120 cell
(600, 1200, 720, 120)
{5, 3, 3}
|
[10]
|
|
|
|
|
Table of sequences
Regular polychoron numbers sequences
N0
|
sequences
|
5
|
{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, ...}
|
8
|
{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, ...}
|
16
|
{0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000, 14641, 20736, 28561, 38416, 50625, 65536, 83521, 104976, 130321, 160000, 194481, 234256, 279841, 331776, 390625, ...}
|
24
|
{0, 1, 24, 153, 544, 1425, 3096, 5929, 10368, 16929, 26200, 38841, 55584, 77233, 104664, 138825, 180736, 231489, 292248, 364249, 448800, 547281, 661144, 791913, 941184, ...}
|
120
|
{0, 1, 120, 947, 3652, 9985, 22276, 43435, 76952, 126897, 197920, 295251, 424700, 592657, 806092, 1072555, 1400176, 1797665, 2274312, 2839987, 3505140, 4280801, ...}
|
600
|
{0, 1, 600, 4983, 19468, 53505, 119676, 233695, 414408, 683793, 1066960, 1592151, 2290740, 3197233, 4349268, 5787615, 7556176, 9701985, 12275208, 15329143, 18920220, ...}
|
See also
Centered regular polychoron numbers
Notes
- ↑ Weisstein, Eric W., Regular Polychoron, From MathWorld--A Wolfram Web Resource.
- ↑ Where
is the d-dimensional regular convex polytope number with N0 0-dimensional elements (vertices V.)
- ↑ Weisstein, Eric W., Polyhedral Formula, From MathWorld--A Wolfram Web Resource.
- ↑ 4.0 4.1 Sondow, Jonathan and Weisstein, Eric W., Riemann Zeta Function, From MathWorld--A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Fermat's Polygonal Number Theorem, From MathWorld--A Wolfram Web Resource.
- ↑ 6.0 6.1 Weisstein, Eric W., Schläfli Symbol, From MathWorld--A Wolfram Web Resource.
- ↑ 7.0 7.1 Weisstein, Eric W., Rising Factorial, From MathWorld--A Wolfram Web Resource.
- ↑ Downey, Lawrence M., Ong, Boon W., and Sellers, James A., Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, 2008.
- ↑ PSYCHEDELIC GEOMETRY, INVERSE POLYGONAL NUMBERS SERIES.
- ↑ 10.0 10.1 10.2 10.3 10.4 HYUN KWANG KIM, ON REGULAR POLYTOPE NUMBERS.
- ↑ Weisstein, Eric W., Multichoose, From MathWorld--A Wolfram Web Resource.
- ↑ Where
, k ≥ 1, n ≥ 0, is the d-dimensional, d ≥ 0, (k+2)-gonal base (hyper)pyramidal number where, for d ≥ 2,
is the number of vertices (including the
apex vertices) of the polygonal base (hyper)pyramid.
- ↑ Weisstein, Eric W., Rhombic Dodecahedral Number, From MathWorld--A 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, 2nd ed., 1994.