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# Regular polychoron numbers

(Redirected from Hypericosahedral numbers)

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]

${\displaystyle P_{N_{0}}^{(4)}(n)={\binom {n+3}{4}}+[N_{0}-5]{\binom {n+2}{4}}+[???]{\binom {n+1}{4}}+[???]{\binom {n}{4}},\,}$

For the 5-cell numbers:

${\displaystyle P_{5}^{(4)}(n)={\binom {n+3}{4}}+[N_{0}-5]{\binom {n+2}{4}}+[N_{0}-5]{\binom {n+1}{4}}+[0]{\binom {n}{4}},\,}$

For the 16-cell and 8-cell numbers:

${\displaystyle P_{8}^{(4)}(n)={\binom {n+3}{4}}+[N_{0}-5]{\binom {n+2}{4}}+[N_{0}-5]{\binom {n+1}{4}}+[1]{\binom {n}{4}},\,}$
${\displaystyle P_{16}^{(4)}(n)={\binom {n+3}{4}}+[N_{0}-5]{\binom {n+2}{4}}+[N_{0}-5]{\binom {n+1}{4}}+[1]{\binom {n}{4}},\,}$

For the 24-cell numbers:

${\displaystyle P_{24}^{(4)}(n)={\binom {n+3}{4}}+[N_{0}-5]{\binom {n+2}{4}}+[43]{\binom {n+1}{4}}+[9]{\binom {n}{4}},\,}$

For the 600-cell and 120-cell numbers:

${\displaystyle P_{120}^{(4)}(n)={\binom {n+3}{4}}+[N_{0}-5]{\binom {n+2}{4}}+[357]{\binom {n+1}{4}}+[107]{\binom {n}{4}},\,}$
${\displaystyle P_{600}^{(4)}(n)={\binom {n+3}{4}}+[N_{0}-5]{\binom {n+2}{4}}+[1993]{\binom {n+1}{4}}+[543]{\binom {n}{4}},\,}$

## Schläfli-Poincaré (convex) polytope formula

Schläfli-Poincaré generalization of the Descartes-Euler (convex) polyhedral formula.[3]

For 4-dimensional (d = 4) regular convex polytopes:

${\displaystyle {\sum _{i=0}^{d-1}(-1)^{i}N_{i}}=N_{0}-N_{1}+N_{2}-N_{3}=V-E+F-C=0,\,}$

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):

${\displaystyle P_{N_{0}}^{(4)}(n)=5P_{N_{0}}^{(4)}(n-1)-10P_{N_{0}}^{(4)}(n-2)+10P_{N_{0}}^{(4)}(n-3)-5P_{N_{0}}^{(4)}(n-4)+P_{N_{0}}^{(4)}(n-5)\,\,}$

with initial conditions

${\displaystyle P_{N_{0}}^{(4)}(0)=?\,}$

For the 5-cell numbers:

${\displaystyle P_{5}^{(4)}(n)=,\,}$

For the 16-cell and 8-cell numbers:

${\displaystyle P_{8}^{(4)}(n)=,\,}$
${\displaystyle P_{16}^{(4)}(n)=,\,}$

For the 24-cell numbers:

${\displaystyle P_{24}^{(4)}(n)=5P_{24}^{(4)}(n-1)-10P_{24}^{(4)}(n-2)+10P_{24}^{(4)}(n-3)-5P_{24}^{(4)}(n-4)+P_{24}^{(4)}(n-5),\,}$

For the 600-cell and 120-cell numbers:

${\displaystyle P_{120}^{(4)}(n)=5P_{120}^{(4)}(n-1)-10P_{120}^{(4)}(n-2)+10P_{120}^{(4)}(n-3)-5P_{120}^{(4)}(n-4)+P_{120}^{(4)}(n-5),\,}$
${\displaystyle P_{600}^{(4)}(n)=5P_{600}^{(4)}(n-1)-10P_{600}^{(4)}(n-2)+10P_{600}^{(4)}(n-3)-5P_{600}^{(4)}(n-4)+P_{600}^{(4)}(n-5).\,}$

## Ordinary generating function

${\displaystyle G_{\{P_{N_{0}}^{(4)}(n)\}}(x)={\frac {x(1+[N_{0}-5]x+[???]x^{2}+[???]x^{3})}{(1-x)^{5}}}\,}$

For the 5-cell numbers:

${\displaystyle G_{\{P_{5}^{(4)}(n)\}}(x)={\frac {x}{(1-x)^{5}}}={\frac {x(1+[N_{0}-5]x+[N_{0}-5]x^{2}+0x^{3})}{(1-x)^{5}}},\,}$

For the 16-cell and 8-cell numbers:

${\displaystyle G_{\{P_{8}^{(4)}(n)\}}(x)={\frac {x(1+3x+3x^{2}+x^{3})}{(1-x)^{5}}}={\frac {x(1+[N_{0}-5]x+[N_{0}-5]x^{2}+x^{3})}{(1-x)^{5}}},\,}$
${\displaystyle G_{\{P_{16}^{(4)}(n)\}}(x)={\frac {x(1+11x+11x^{2}+x^{3})}{(1-x)^{5}}}={\frac {x(1+[N_{0}-5]x+[N_{0}-5]x^{2}+x^{3})}{(1-x)^{5}}},\,}$

For the 24-cell numbers:

${\displaystyle G_{\{P_{24}^{(4)}(n)\}}(x)={\frac {x(9x^{3}+43x^{2}+19x+1)}{(1-x)^{5}}}={\frac {x(1+[N_{0}-5]x+43x^{2}+9x^{3})}{(1-x)^{5}}},\,}$

For the 600-cell and 120-cell numbers:

${\displaystyle G_{\{P_{120}^{(4)}(n)\}}(x)={\frac {x(107x^{3}+357x^{2}+115x+1)}{(1-x)^{5}}}={\frac {x(1+[N_{0}-5]x+357x^{2}+107x^{3})}{(1-x)^{5}}},\,}$
${\displaystyle G_{\{P_{600}^{(4)}(n)\}}(x)={\frac {x(543x^{3}+1993x^{2}+595x+1)}{(1-x)^{5}}}={\frac {x(1+[N_{0}-5]x+1993x^{2}+543x^{3})}{(1-x)^{5}}},\,}$

### Procedure to obtain the generating functions

${\displaystyle G_{\{P_{24}^{(4)}(n)\}}(x)=G_{\{n^{2}(3n^{2}-4n+2)\}}(x)=G_{\{3n^{4}-4n^{3}+2n^{2}\}}(x)=3G_{\{n^{4}\}}(x)-4G_{\{n^{3}\}}(x)+2G_{\{n^{2}\}}(x)\,}$
${\displaystyle ={\frac {x}{(1-x)^{5}}}\{3(1+11x+11x^{2}+x^{3})-4(1-x)(1+4x+x^{2})+2(1-x)^{2}(1+x)\}={\frac {x(9x^{3}+43x^{2}+19x+1)}{(1-x)^{5}}},\,}$
${\displaystyle G_{\{P_{120}^{(4)}(n)\}}(x)=G_{\{{\frac {n(145n^{3}-280n^{2}+179n-38)}{6}}\}}(x)={\frac {1}{6}}G_{\{145n^{4}-280n^{3}+179n^{2}-38n\}}(x)={\frac {1}{6}}\{145G_{\{n^{4}\}}(x)-280G_{\{n^{3}\}}(x)+179G_{\{n^{2}\}}(x)-38G_{\{n\}}(x)\}\,}$
${\displaystyle ={\frac {x}{6(1-x)^{5}}}\{145(1+11x+11x^{2}+x^{3})-280(1-x)(1+4x+x^{2})+179(1-x)^{2}(1+x)-38(1-x)^{3}\}={\frac {x(107x^{3}+357x^{2}+115x+1)}{(1-x)^{5}}},\,}$
${\displaystyle G_{\{P_{600}^{(4)}(n)\}}(x)=G_{\{{\frac {n(261n^{3}-504n^{2}+283n-38)}{2}}\}}(x)={\frac {1}{2}}G_{\{261n^{4}-504n^{3}+283n^{2}-38n\}}(x)={\frac {1}{2}}\{261G_{\{n^{4}\}}(x)-504G_{\{n^{3}\}}(x)+283G_{\{n^{2}\}}(x)-38G_{\{n\}}(x)\}\,}$
${\displaystyle ={\frac {x}{2(1-x)^{5}}}\{261(1+11x+11x^{2}+x^{3})-504(1-x)(1+4x+x^{2})+283(1-x)^{2}(1+x)-38(1-x)^{3}\}={\frac {x(543x^{3}+1993x^{2}+595x+1)}{(1-x)^{5}}},\,}$
${\displaystyle G_{\{n^{4}\}}(x)={\frac {x\ E_{4}(x)}{(1-x)^{5}}}={\frac {x\ (1+11x+11x^{2}+x^{3})}{(1-x)^{5}}},\,}$
${\displaystyle G_{\{n^{3}\}}(x)={\frac {x\ E_{3}(x)}{(1-x)^{4}}}={\frac {x\ (1+4x+x^{2})}{(1-x)^{4}}},\,}$
${\displaystyle G_{\{n^{2}\}}(x)={\frac {x\ E_{2}(x)}{(1-x)^{3}}}={\frac {x\ (1+x)}{(1-x)^{3}}},\,}$
${\displaystyle G_{\{n\}}(x)={\frac {x\ E_{1}(x)}{(1-x)^{2}}}={\frac {x\ (1)}{(1-x)^{2}}}.\,}$

## Exponential generating function

${\displaystyle E_{\{P_{N_{0}}^{(4)}(n)\}}(x)=?\,}$

For the 5-cell numbers:

${\displaystyle E_{\{P_{5}^{(4)}(n)\}}(x)=,\,}$

For the 16-cell and 8-cell numbers:

${\displaystyle E_{\{P_{8}^{(4)}(n)\}}(x)=,\,}$
${\displaystyle E_{\{P_{16}^{(4)}(n)\}}(x)=(x+7x^{2}+6x^{3}+x^{4})\ e^{x},\,}$

For the 24-cell numbers:

${\displaystyle E_{\{P_{24}^{(4)}(n)\}}(x)=,\,}$

For the 600-cell and 120-cell numbers:

${\displaystyle E_{\{P_{120}^{(4)}(n)\}}(x)=,\,}$
${\displaystyle E_{\{P_{600}^{(4)}(n)\}}(x)=,\,}$

## Dirichlet generating function

${\displaystyle D_{\{P_{N_{0}}^{(4)}(n)\}}(x)=?\,}$

For the 5-cell numbers:

${\displaystyle D_{\{P_{5}^{(4)}(n)\}}(x)=,\,}$

For the 16-cell and 8-cell numbers:

${\displaystyle D_{\{P_{8}^{(4)}(n)\}}(x)=,\,}$
${\displaystyle D_{\{P_{16}^{(4)}(n)\}}(x)=\zeta (s-4),\,}$ [4]

For the 24-cell numbers:

${\displaystyle D_{\{P_{24}^{(4)}(n)\}}(x)=,\,}$

For the 600-cell and 120-cell numbers:

${\displaystyle D_{\{P_{120}^{(4)}(n)\}}(x)=,\,}$
${\displaystyle D_{\{P_{600}^{(4)}(n)\}}(x)=,\,}$

## 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 ${\displaystyle \scriptstyle A\,}$ of nonnegative integers is called a basis of order ${\displaystyle \scriptstyle g\,}$ if ${\displaystyle \scriptstyle g\,}$ is the minimum number with the property that every nonnegative integer can be written as a sum of ${\displaystyle \scriptstyle g\,}$ elements in ${\displaystyle \scriptstyle A\,}$. Lagrange’s sum of four squares can be restated as the set ${\displaystyle \scriptstyle \{n^{2}|n=0,1,2,\ldots \}\,}$ of nonnegative squares forms a basis of order 4.

Theorem (Cauchy) For every ${\displaystyle \scriptstyle k\geq 3}$, the set ${\displaystyle \scriptstyle \{P(k,n)|n=0,1,2,\ldots \}\,}$ of k-gon numbers forms a basis of order ${\displaystyle \scriptstyle k\,}$, i.e. every nonnegative integer can be written as a sum of ${\displaystyle \scriptstyle k\,}$ 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 ${\displaystyle \scriptstyle g(d)\,}$ such that every nonnegative integer is a sum of ${\displaystyle \scriptstyle g(d)\,}$ ${\displaystyle \scriptstyle d\,}$th powers, i.e. the set ${\displaystyle \scriptstyle \{n^{d}|n=0,1,2,\ldots \}\,}$ of ${\displaystyle \scriptstyle d\,}$th powers forms a basis of order ${\displaystyle \scriptstyle g(d)\,}$. The Hilbert-Waring problem is concerned with the study of ${\displaystyle \scriptstyle g(d)\,}$ for ${\displaystyle \scriptstyle d\geq 2\,}$. 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

${\displaystyle P_{N_{0}}^{(4)}(n)-P_{N_{0}}^{(4)}(n-1)=?,\,}$

## Partial sums

The partial sums correspond to 5-dimensional polychoron hyperpyramidal numbers.

${\displaystyle \sum _{n=1}^{m}P_{N_{0}}^{(4)}(n)=?,\,}$

## Partial sums of reciprocals

${\displaystyle \sum _{n=1}^{m}{\frac {1}{P_{N_{0}}^{(4)}(n)}}=?,\,}$

## Sum of reciprocals

${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{P_{N_{0}}^{(4)}(n)}}=?,\,}$

## 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

${\displaystyle P_{N_{0}}^{(4)}(n)=\,}$

${\displaystyle \scriptstyle {\binom {n+3}{4}}+[N_{0}-5]{\binom {n+2}{4}}+[???]{\binom {n+1}{4}}+[???]{\binom {n}{4}},\,}$

Generating

function

${\displaystyle G_{\{P_{N_{0}}^{(4)}(n)\}}(x)=\,}$

${\displaystyle \scriptstyle {{x(1+??x+??x^{2}+??x^{3})} \over {(1-x)^{5}}}\,}$

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}

${\displaystyle {\binom {n+3}{4}}}$

${\displaystyle {\frac {n^{(4)}}{4!}}}$[7]

${\displaystyle {{(n+3)P_{4}^{(3)}(n)} \over 4}}$

${\displaystyle x \over (1-x)^{5}\,}$ 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}

${\displaystyle \scriptstyle {\binom {n+3}{4}}+3{\binom {n+2}{4}}+3{\binom {n+1}{4}}+{\binom {n}{4}}\,}$

${\displaystyle {n^{2}(n^{2}+2) \over 3}\,}$

${\displaystyle \scriptstyle {\frac {x(1+3x+3x^{2}+x^{3})}{(1-x)^{5}}}\,}$

${\displaystyle {\frac {x(1+x)^{3}}{(1-x)^{5}}}\,}$

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}

${\displaystyle \scriptstyle {\binom {n+3}{4}}+11{\binom {n+2}{4}}+11{\binom {n+1}{4}}+{\binom {n}{4}}\,}$

${\displaystyle n^{4}\,}$

${\displaystyle \scriptstyle {{x(1+11x+11x^{2}+x^{3})} \over {(1-x)^{5}}}\,}$ 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}

${\displaystyle \scriptstyle {\binom {n+3}{4}}+19{\binom {n+2}{4}}+43{\binom {n+1}{4}}+9{\binom {n}{4}}\,}$

${\displaystyle n^{2}(3n^{2}-4n+2)\,}$

${\displaystyle \scriptstyle {{x(1+19x+43x^{2}+9x^{3})} \over {(1-x)^{5}}}\,}$ 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}

${\displaystyle \scriptstyle {\binom {n+3}{4}}+115{\binom {n+2}{4}}+357{\binom {n+1}{4}}+107{\binom {n}{4}}\,}$

${\displaystyle \scriptstyle {{n(145n^{3}-280n^{2}+179n-38)} \over 6}\,}$

${\displaystyle \scriptstyle {{x(1+115x+357x^{2}+107x^{3})} \over {(1-x)^{5}}}\,}$ 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}

${\displaystyle \scriptstyle {\binom {n+3}{4}}+595{\binom {n+2}{4}}+1993{\binom {n+1}{4}}+543{\binom {n}{4}}\,}$

${\displaystyle \scriptstyle {{n(261n^{3}-504n^{2}+283n-38)} \over 2}\,}$

${\displaystyle \scriptstyle {{x(1+595x+1993x^{2}+543x^{3})} \over {(1-x)^{5}}}\,}$ 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

${\displaystyle g_{\{P_{N_{0}}^{(4)}\}}=\,}$

${\displaystyle N_{0}+?\,}$

Differences

${\displaystyle P_{N_{0}}^{(4)}(n)-\,}$

${\displaystyle P_{N_{0}}^{(4)}(n-1)=\,}$

Partial sums

${\displaystyle \sum _{n=1}^{m}{P_{N_{0}}^{(4)}(n)}}$

Partial sums of reciprocals

${\displaystyle \sum _{n=1}^{m}{1 \over {P_{N_{0}}^{(4)}(n)}}}$

Sum of Reciprocals[8][9]

${\displaystyle \sum _{n=1}^{\infty }{1 \over {P_{N_{0}}^{(4)}}}}$

0 5

${\displaystyle \scriptstyle 2^{r+2}+0^{r}\,}$

Pentachoron

5 cell

(5, 10, 10, 5)

{3, 3, 3}

${\displaystyle 8?\,}$

${\displaystyle (N_{0}+3?)\,}$[10]

${\displaystyle P_{3+1}^{(3)}(n)\,}$

(A000292(n))

${\displaystyle {\binom {m+4}{5}}\,}$

${\displaystyle {\frac {m^{(5)}}{5!}}\,}$ [7]

${\displaystyle \left(\!\!{\binom {m}{5}}\!\!\right)\,}$ [11]

${\displaystyle \scriptstyle {\frac {m(m+1)(m+2)(m+3)(m+4)}{120}}\,}$

${\displaystyle \,}$ ${\displaystyle {\frac {4}{3}}}$
1 8

${\displaystyle \scriptstyle 2^{r+2}+0^{r}\,}$

16 cell

(8, 24, 32, 16)

{3, 3, 4}

${\displaystyle 11?\,}$

${\displaystyle (N_{0}+3?)\,}$[10]

${\displaystyle {\,}_{c}P_{2\cdot 3}^{(3)}(n)\,}$

(A001845(n-1))

${\displaystyle {\binom {2m+2}{3}}\ {\frac {m^{2}+m+3}{20}}\,}$

${\displaystyle {\frac {(2m)^{(3)}\ (m^{2}+m+3)}{5!}}\,}$

${\displaystyle \scriptstyle {\frac {m(m+1)(2m+1)(m^{2}+m+3)}{30}}\,}$

A061927(n-1)?

${\displaystyle \,}$ ${\displaystyle \,}$
2 16

${\displaystyle \scriptstyle 2^{r+2}+0^{r}\,}$

Tesseract

8 cell

(16, 32, 24, 8)

{4, 3, 3}

${\displaystyle 19\,}$

${\displaystyle (N_{0}+3)\,}$

${\displaystyle {\,}_{c}P_{2^{3}}^{(3)}(n)+6\ Y_{5}^{(3)}(n-1)\,}$ [12]

Centered cubic(n) +
6 Square pyramidal(n-1)

(A005917(n-1))

${\displaystyle {\binom {2m+2}{3}}\ {\frac {3m^{2}+3m-1}{20}}\,}$

${\displaystyle \scriptstyle {\frac {(2m)^{(3)}\ (3m^{2}+3m-1)}{5!}}\,}$

${\displaystyle \scriptstyle {\frac {m(m+1)(2m+1)(3m^{2}+3m-1)}{30}}\,}$

${\displaystyle \,}$ ${\displaystyle \zeta (4)={{\pi ^{4}} \over 90}\,}$[4]
3 24

${\displaystyle \scriptstyle (r+1)!\,}$

${\displaystyle \scriptstyle (r+1)\cdot r!\,}$

24 cell

(24, 96, 96, 24)

{3, 4, 3}

${\displaystyle 28?\,}$

${\displaystyle (N_{0}+4?)\,}$[10]

${\displaystyle \scriptstyle 12n^{3}-30n^{2}+28n-9\,}$ ${\displaystyle \scriptstyle {\binom {m+1}{2}}\ {\frac {18m^{3}-3m^{2}-7m+7}{15}}\,}$

${\displaystyle \scriptstyle {\frac {m(m+1)(18m^{3}-3m^{2}-7m+7)}{30}}\,}$

${\displaystyle \,}$ ${\displaystyle \,}$
4 120

${\displaystyle \scriptstyle (r+1)!\,}$

${\displaystyle \scriptstyle (r+1)\cdot r!\,}$

600 cell

(120, 720, 1200, 600)

{3, 3, 5}

${\displaystyle 125?\,}$

${\displaystyle (N_{0}+5?)\,}$[10]

${\displaystyle \scriptstyle {\frac {290n^{3}-855n^{2}+889n-321}{3}}\,}$ ${\displaystyle \scriptstyle {{\binom {m+1}{2}}\ {\frac {58m^{3}-53m^{2}-11m+12}{6}}}\,}$

${\displaystyle \scriptstyle {\frac {m(m+1)(58m^{3}-53m^{2}-11m+12)}{12}}\,}$

${\displaystyle \,}$ ${\displaystyle \,}$
5 600

${\displaystyle \scriptstyle r\cdot r!\,}$

120 cell

(600, 1200, 720, 120)

{5, 3, 3}

${\displaystyle 606?\,}$

${\displaystyle (N_{0}+6?)\,}$[10]

${\displaystyle \scriptstyle {522n^{3}-1539n^{2}+1561n-543}\,}$ ${\displaystyle \scriptstyle {{\binom {m+1}{2}}\ {\frac {1566m^{3}-1431m^{2}-689m+584}{30}}}\,}$

${\displaystyle \scriptstyle {\frac {m(m+1)(1566m^{3}-1431m^{2}-689m+584)}{60}}\,}$

${\displaystyle \,}$ ${\displaystyle \,}$

## Table of sequences

Regular polychoron numbers sequences
N0 ${\displaystyle P_{N_{0}}^{(4)}(n),\ n\geq 0}$ 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, ...}

## Notes

1. Weisstein, Eric W., Regular Polychoron, From MathWorld--A Wolfram Web Resource.
2. Where ${\displaystyle \scriptstyle P_{N_{0}}^{(d)}(n)\,}$ is the d-dimensional regular convex polytope number with N0 0-dimensional elements (vertices V.)
3. Weisstein, Eric W., Polyhedral Formula, From MathWorld--A Wolfram Web Resource.
4. Sondow, Jonathan and Weisstein, Eric W., Riemann Zeta Function, From MathWorld--A Wolfram Web Resource.
5. Weisstein, Eric W., Fermat's Polygonal Number Theorem, From MathWorld--A Wolfram Web Resource.
6. Weisstein, Eric W., Schläfli Symbol, From MathWorld--A Wolfram Web Resource.
7. Weisstein, Eric W., Rising Factorial, From MathWorld--A Wolfram Web Resource.
8. Downey, Lawrence M., Ong, Boon W., and Sellers, James A., Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, 2008.
9. PSYCHEDELIC GEOMETRY, INVERSE POLYGONAL NUMBERS SERIES.
10. HYUN KWANG KIM, ON REGULAR POLYTOPE NUMBERS.
11. Weisstein, Eric W., Multichoose, From MathWorld--A Wolfram Web Resource.
12. Where ${\displaystyle \scriptstyle Y_{[(k+2)+(d-2)]}^{(d)}(n)=Y_{k+d}^{(d)}(n)\,}$, k ≥ 1, n ≥ 0, is the d-dimensional, d ≥ 0, (k+2)-gonal base (hyper)pyramidal number where, for d ≥ 2, ${\displaystyle \scriptstyle N_{0}=[(k+2)+(d-2)]\,}$ is the number of vertices (including the ${\displaystyle \scriptstyle d-2\,}$ apex vertices) of the polygonal base (hyper)pyramid.
13. Weisstein, Eric W., Rhombic Dodecahedral Number, From MathWorld--A Wolfram Web Resource.