Regular orthotopic numbers

The regular orthotopic numbers are a family of sequences of figurate numbers corresponding to the

d

-dimensional regular orthotope for each dimension

d

, where

d

is a nonnegative integer. These include the square numbers, the cube numbers and the hypercube numbers for

d > 3

.

The

d

-dimensional regular orthotopic numbers, forming regular orthotopes (e.g. points, segments, squares, cubes and then hypercubes), where (-1)-cells correspond to the null polytope, 0-cells are vertices, 1-cells are edges, 2-cells are faces, and so on...

d = 0	Regular 0-orthotopic numbers	Point numbers	Form point	(0 (-1)-cell facets)	(regular 0-orthotope)
d = 1	Regular 1-orthotopic numbers	Linear numbers	Form segments	(2 0-cell facets)	(regular 1-orthotope)
d = 2	Regular 2-orthotopic numbers	Square numbers (tetragonal numbers)	Form squares	(4 1-cell facets)	(regular 2-orthotope)
d = 3	Regular 3-orthotopic numbers	Cube numbers (hexahedral numbers)	Form cubes	(6 2-cell facets)	(regular 3-orthotope)
d = 4	Regular 4-orthotopic numbers	Octachoron numbers	Form tesseracts	(8 3-cell facets)	(regular 4-orthotope)
d = 5	Regular 5-orthotopic numbers	Decateron numbers	Form penteracts	(10 4-cell facets)	(regular 5-orthotope)
d = 6	Regular 6-orthotopic numbers	Dodecapeton numbers	Form hexeracts	(12 5-cell facets)	(regular 6-orthotope)
d = 7	Regular 7-orthotopic numbers	Tetradecahexon numbers	Form hepteracts	(14 6-cell facets)	(regular 7-orthotope)
d = 8	Regular 8-orthotopic numbers	Hexadecahepton numbers	Form octeracts	(16 7-cell facets)	(regular 8-orthotope)
...	...	...	...	...	...
d = d	Regular d-orthotopic numbers	(2d) (d-1)-cell numbers	Form (d)-eracts	(2d (d-1)-cell facets)	(regular d-orthotope)

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

Formulae

The

n

^th regular

d

-orthotopic numbers are given by the formulae

$P_{2^{d}}^{(d)}(n)=n^{d},\,$

where $\scriptstyle d\,\geq \,0\,$ is the dimension and $\scriptstyle n-1\,$ is the number of nondegenerate layered regular orthotopes ($\scriptstyle n\,=\,0\,$ giving no dot and $\scriptstyle n\,=\,1\,$ giving a single dot, a degenerate regular orthotope) of the

d

-dimensional regular orthotopic number (regular

d

-orthotope number.)

Recurrence relation

$P_{2^{d}}^{(d)}(n)=P_{2^{d}}^{(d)}(n-1)+\sum _{i=1}^{d}{(-1)}^{i-1}{\binom {d}{i}}n^{d-i}\,$

Generating function

$G_{\{n^{d}\}}(x)={\frac {x\ A_{d}(x)}{(1-x)^{d+1}}}\,$

where

$A_{d}(x)=\sum _{k=1}^{d}A(d,k)\ x^{k-1}\,$

is the

d

^th Eulerian polynomial (Cf. Talk:Regular_orthotopic_numbers) whose Eulerian numbers

$A(d,k)|_{k=1}^{d}=\{1,(2^{d}-d-1),...,(2^{d}-d-1),1\}\,$

can be recursively generated with the triangle of Eulerian numbers.

$G_{\{P_{2^{d}}^{(d)}(n)\}}(x)=G_{\{n^{d}\}}(x)={\frac {x}{(1-x)}}\sum _{i=0}^{d-1}{\binom {d}{i}}G_{\{n^{i}\}}(x),\,$ with $G_{\{n^{0}\}}(x)=G_{\{1\}}(x)={\frac {1}{1-x}}\,$

$G_{\{P_{2^{d}}^{(d)}(n)\}}(x)={\frac {x(1+(2^{d}-d-1)x+...+(2^{d}-d-1)x^{d-2}+x^{d-1})}{(1-x)^{d+1}}}\,$

$G_{\{P_{2^{d}}^{(d)}(n)\}}(x)=Li_{-d}(x)\,$

Method for obtaining the generating functions for successive powers

Since $\scriptstyle \sum _{i=0}^{\infty }1x^{i}\,=\,{\frac {1}{1-x}},\ |x|\,<\,1\,$, the generating function of 1 is then

$G_{\{1\}}(x)={\frac {1}{1-x}}.\,$

Since $\scriptstyle G_{\{n+1\}}(x)\,=\,{\frac {G_{\{n\}}(x)}{x}}\,=\,G_{\{n\}}(x)+G_{\{1\}}(x)\,$, the generating function of $\scriptstyle n\,$ is then

$G_{\{n\}}(x)=xG_{\{n\}}(x)+x{\frac {1}{1-x}},\,$ and $(1-x)G_{\{n\}}(x)={\frac {x}{1-x}},\,$ which gives

$G_{\{n\}}(x)={\frac {x}{(1-x)^{2}}}.\,$

Since $\scriptstyle G_{\{(n+1)^{2}\}}(x)\,=\,{\frac {G_{\{n^{2}\}}(x)}{x}}\,=\,G_{\{n^{2}\}}(x)+2G_{\{n\}}(x)+G_{\{1\}}(x)\,$, the generating function of $\scriptstyle n^{2}\,$ is then

$G_{\{n^{2}\}}(x)=xG_{\{n^{2}\}}(x)+2x{\frac {x}{(1-x)^{2}}}+x{\frac {1}{1-x}},\,$ and $(1-x)G_{\{n^{2}\}}(x)={\frac {2x^{2}+x(1-x)}{(1-x)^{2}}}={\frac {x(1+x)}{(1-x)^{2}}},\,$ which gives

$G_{\{n^{2}\}}(x)={\frac {x(1+x)}{(1-x)^{3}}}.\,$

Since $\scriptstyle G_{\{(n+1)^{3}\}}(x)\,=\,{\frac {G_{\{n^{3}\}}(x)}{x}}\,=\,G_{\{n^{3}\}}(x)+3G_{\{n^{2}\}}(x)+3G_{\{n\}}(x)+G_{\{1\}}(x)\,$, the generating function of $\scriptstyle n^{3}\,$ is then

$G_{\{n^{3}\}}(x)=xG_{\{n^{3}\}}(x)+3x{\frac {x(1+x)}{(1-x)^{3}}}+3x{\frac {x}{(1-x)^{2}}}+x{\frac {1}{1-x}},\,$ and $(1-x)G_{\{n^{3}\}}(x)={\frac {3x^{2}(1+x)+3x^{2}(1-x)+x(1-x)^{2}}{(1-x)^{3}}}={\frac {x(1+4x+x^{2})}{(1-x)^{3}}},\,$ which gives

$G_{\{n^{3}\}}(x)={\frac {x(1+4x+x^{2})}{(1-x)^{4}}}.\,$

Since $\scriptstyle G_{\{(n+1)^{4}\}}(x)\,=\,{\frac {G_{\{n^{4}\}}(x)}{x}}\,=\,G_{\{n^{4}\}}(x)+4G_{\{n^{3}\}}(x)+6G_{\{n^{2}\}}(x)+4G_{\{n\}}(x)+G_{\{1\}}(x)\,$, the generating function of $\scriptstyle n^{4}\,$ is then

$G_{\{n^{4}\}}(x)=xG_{\{n^{4}\}}(x)+4x{\frac {x(1+4x+x^{2})}{(1-x)^{4}}}+6x{\frac {x(1+x)}{(1-x)^{3}}}+4x{\frac {x}{(1-x)^{2}}}+x{\frac {1}{1-x}},\,$ and $(1-x)G_{\{n^{4}\}}(x)={\frac {4x^{2}(1+4x+x^{2})+6x^{2}(1+x)(1-x)+4x^{2}(1-x)^{2}+x(1-x)^{3}}{(1-x)^{4}}}={\frac {x(1+11x+11x^{2}+x^{3})}{(1-x)^{4}}},\,$ which gives

$G_{\{n^{4}\}}(x)={\frac {x(1+11x+11x^{2}+x^{3})}{(1-x)^{5}}}.\,$

Since $\scriptstyle G_{\{(n+1)^{5}\}}(x)\,=\,{\frac {G_{\{n^{5}\}}(x)}{x}}\,=\,G_{\{n^{5}\}}(x)+5G_{\{n^{4}\}}(x)+10G_{\{n^{3}\}}(x)+10G_{\{n^{2}\}}(x)+5G_{\{n\}}(x)+G_{\{1\}}(x)\,$, the generating function of $\scriptstyle n^{5}\,$ is then

$G_{\{n^{5}\}}(x)=xG_{\{n^{5}\}}(x)+...,\,$ and $(1-x)G_{\{n^{5}\}}(x)=...={\frac {x(1+26x+66x^{2}+26x^{3}+x^{4})}{(1-x)^{5}}},\,$ which gives

$G_{\{n^{5}\}}(x)={\frac {x(1+26x+66x^{2}+26x^{3}+x^{4})}{(1-x)^{6}}}.\,$

And the general recurrence equation for the generating function of powers is

$G_{\{n^{d}\}}(x)={\frac {x}{(1-x)}}\sum _{i=0}^{d-1}{\binom {d}{i}}G_{\{n^{i}\}}(x),\,$ with $G_{\{n^{0}\}}(x)=G_{\{1\}}(x)={\frac {1}{1-x}}\,$

Simpler method for obtaining the generating functions for successive powers

Since $\scriptstyle \sum _{i=0}^{\infty }1x^{i}\,=\,{\frac {1}{1-x}},\ |x|\,<\,1\,$, we have

$G_{\{n^{0}\}}(x)={\frac {1}{1-x}},\,$

then

$G_{\{n^{d}\}}(x)=x\ {G'}_{\{n^{d-1}\}}(x)=x\ {\frac {dG_{\{n^{d-1}\}}(x)}{dx}},\ d\geq 1.\,$

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-gonal numbers. Fermat claimed to have a proof of this result, although Fermat’s proof has never been found. 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-gonal 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

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 $\scriptstyle \{n^{2}\,|\,n\,=\,0,\,1,\,2,\,\ldots \}\,$ of nonnegative squares forms a basis of order 4. Theorem (Cauchy) For every

k   ≥   3

, the set $\scriptstyle \{P(k,n)\,|\,n\,=\,0,\,1,\,2,\,\ldots \}\,$ of $\scriptstyle k\,$-gonal numbers forms a basis of order

k

, i.e. every nonnegative integer can be written as a sum of

k

k

-gonal 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 $\scriptstyle g(d)\,$ such that every nonnegative integer is a sum of $\scriptstyle g(d)\,$ $\scriptstyle d\,$^th powers, i.e. the set $\scriptstyle \{n^{d}\,|\,n\,=\,0,\,1,\,2,\,\ldots \}\,$ of $\scriptstyle d\,$^th powers forms a basis of order $\scriptstyle g(d)\,$. The Hilbert-Waring problem is concerned with the study of

g (d)

for

d   ≥   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.

The (presumed) solution to Waring’s problem is (see A002804)

$g(n)=(2^{n}-2)+{\bigg \lfloor }{{{\bigg (}{3 \over 2}{\bigg )}}^{n}}{\bigg \rfloor }=2(2^{n-1}-1)+{\bigg \lfloor }{{{\bigg (}{\frac {3}{2}}{\bigg )}}^{n}}{\bigg \rfloor },\,$

with series representation

$g(n)=-{5 \over 2}+{\bigg (}{3 \over 2}{\bigg )}^{n}+2^{n}+{\sum _{k=1}^{\infty }{\sin(({3 \over 2})^{n}2k\pi ) \over k} \over \pi }\,$ for $({(3/2)}^{n}\in \mathbb {R} \,$ and $({3/2)}^{n}\not \in \mathbb {Z} )\,$.

(Presumed) solution to Waring’s problem

n	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
$\scriptstyle (2^{n}-2)+\lfloor {{({3 \over 2})}^{n}}\rfloor \,$	1	4	9	19	37	73	143	279	548	1079	2132	4223	8384	16673	33203

Differences

$P_{2^{d}}^{(d)}(n)-P_{2^{d}}^{(d)}(n-1)=\sum _{k=0}^{d-1}{\binom {d}{k}}(n-1)^{k}\,$

Partial sums

$\sum _{n=1}^{m}P_{2^{d}}^{(d)}(n)=H_{m}^{(-d)}\,$

Partial sums of reciprocals

$\sum _{n=1}^{m}{\frac {1}{P_{2^{d}}^{(d)}(n)}}=H_{m}^{(d)}\,$

Sum of reciprocals

$\sum _{n=1}^{\infty }{\frac {1}{P_{2^{d}}^{(d)}(n)}}=\zeta (d)\,$

Number of j-dimensional “vertices”

$N_{j}=2^{d-j}{\binom {d}{j}},\ (0\leq j\leq d)\,$

Table of formulae and values

N0, N1, N2, N3, ...

are the number of vertices (0-dimensional elements), edges (1-dimensional elements), faces (2-dimensional elements), cells (3-dimensional elements)... respectively, where the (

d  −  1

)-dimensional elements are the actual facets. The regular orthotopic numbers are listed by increasing number

N0

of vertices.

Regular orthotopic numbers formulae and values

d	Name Regular d -orthotope 2d (d  −  1) -cell (  N0, N1, N2, ... ) Schläfli symbol	Formulae P  (d  ) 2 d(n) = n d	n = 0	1	2	3	4	5	6	7	8	9	10	11	12	A-number
0	Point numbers 0-orthotope zero-(-1)-cell () {}	n 0, n   ≥   1 0, n = 0.	0	1	1	1	1	1	1	1	1	1	1	1	1	(NOT A057427)
1	Segment numbers 1-orthotope di-0-cell (2) {}	n	0	1	2	3	4	5	6	7	8	9	10	11	12	A001477
2	Square numbers 2-orthotope tetra-1-cell (4, 4) {4}	n 2	0	1	4	9	16	25	36	49	64	81	100	121	144	A000290
3	Cube numbers 3-orthotope hexa-2-cell (8, 12, 6) {4, 3}	n 3	0	1	8	27	64	125	216	343	512	729	1000	1331	1728	A000578
4	Tesseract numbers 4-orthotope octa-3-cell (16, 32, 24, 8) {4, 3, 3}	n 4	0	1	16	81	256	625	296	2401	4096	6561	10000	14641	20736	A000583
5	Penteract numbers 5-orthotope deca-4-cell (32, 80, 80, 40, 10) {4, 3, 3, 3}	n 5	0	1	32	243	1024	3125	7776	16807	32768	59049	100000	161051	248832	A000584
6	Hexeract numbers 6-orthotope dodeca-5-cell (64, 192, 240, 160, 60, 12) {4, 3, 3, 3, 3}	n 6	0	1	64	729	4096	15625	46656	117649	262144	531441	1000000	1771561	2985984	A001014
7	Hepteract numbers 7-orthotope tetradeca-6-cell (128, 448, 672, 560, 280, 84, 14) {4, 3, 3, 3, 3, 3}	n 7	0	1	128	2187	16384	78125	279936	823543	2097152	4782969	10000000	19487171	35831808	A001015
8	Octeract numbers 8-orthotope hexadeca-7-cell (256, 1024, 1792, 1792, 1120, 448, 112, 16) {4, 3, 3, 3, 3, 3, 3}	n 8	0	1	256	6561	65536	390625	1679616	5764801	16777216	43046721	100000000	214358881	429981696	A001016
9	Enneract numbers 9-orthotope octadeca-8-cell (512, 2304, 4608, 5376, 4032, 2016, 672, 144, 18) {4, 3, 3, 3, 3, 3, 3, 3}	n 9	0	1	512	19683	262144	1953125	10077696	40353607	134217728	387420489	1000000000	2357947691	5159780352	A001017
10	Dekeract numbers 10-orthotope icosa-9-cell (1024, 5120, 11520, 15360, 13440, 8064, 3360, 960, 180, 20) {4, 3, 3, 3, 3, 3, 3, 3, 3}	n 10	0	1	1024	59049	1048576	9765625	60466176	282475249	1073741824	3486784401	10000000000	25937424601	61917364224	A008454
11	Hendekeract numbers 11-orthotope icosidi-10-cell (..., 22) {4, 3, 3, 3, 3, 3, 3, 3, 3, 3}	n 11	0	1	2048	177147	4194304	48828125	362797056	1977326743	8589934592	31381059609	100000000000	285311670611	743008370688	A008455
12	Dodekeract numbers 12-orthotope icositetra-11-cell (..., 24) {4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3}	n 12	0	1	4096	531441	16777216	244140625	2176782336	13841287201	68719476736	282429536481	1000000000000	3138428376721	8916100448256	A008456

Table of related formulae and values

N0, N1, N2, N3, ...

are the number of vertices (0-dimensional elements), edges (1-dimensional elements), faces (2-dimensional elements), cells (3-dimensional elements)... respectively, where the (

d  −  1

)-dimensional elements are the actual facets. The regular orthotopic numbers are listed by increasing number

N0

of vertices.

Regular orthotopic numbers related formulae and values

$d\,$	Name Regular $\scriptstyle d\,$-orthotope $\scriptstyle 2d\ (d-1)\,$-cell $\scriptstyle (N_{0},\,N_{1},\,N_{2},\,\ldots )\,$ Schläfli symbol	Generating function $G_{\{P_{2^{d}}^{(d)}(n)\}}(x)=\,$ ${\frac {x\ A_{d}(x)}{(1-x)^{n+1}}},\,$ where $\scriptstyle A_{d}(x)\,$ is the $\scriptstyle d\,$th Eulerian polynomial (Cf. Talk:Regular_orthotopic_numbers) given by $\scriptstyle A_{d}(x)\,=\,\sum _{k=1}^{d}A(d,k)\ x^{k-1},\,$ where $\scriptstyle A(d,k)|_{k=1}^{d}\,=\,\{1,(2^{d}-d-1),\dots ,(2^{d}-d-1),1\}\,$ are the Eulerian numbers. (Cf. triangle of Eulerian numbers) $\scriptstyle {\frac {x}{(1-x)}}\sum _{i=0}^{d-1}{\binom {d}{i}}G(x^{i}),\ G(x^{0})\,=\,G(1)\,=\,{\frac {1}{1-x}}\,$	Order of basis $g_{\{P_{2^{d}}^{(d)}\}}\,$ A002804	Differences $P_{2^{d}}^{(d)}(n)-\,$ $P_{2^{d}}^{(d)}(n-1)=\,$ $\sum _{k=0}^{d-1}{\binom {d}{k}}(n-1)^{k}\,$	Partial sums $\sum _{n=1}^{m}{P_{2^{d}}^{(d)}(n)}=\,$ $H_{m}^{(-d)}\,$	Partial sums of reciprocals $\sum _{n=1}^{m}{1 \over {P_{2^{d}}^{(d)}(n)}}=$ $H_{m}^{(d)}\,$	Sum of reciprocals $\sum _{n=1}^{\infty }{1 \over {P_{2^{d}}^{(d)}(n)}}=$ $\zeta (d),\,$ $\zeta (2k)=\,$ ${{2^{2k-1}|B_{2k}|\pi ^{2k}} \over {(2k)!}},\,$ $(k\in \mathbb {N} ^{+})\,$
0	Point numbers 0-orthotope zero-(-1)-cell () {}	${\frac {x\ A_{0}(x)}{(1-x)}},\,$ with $\scriptstyle A_{0}(x)=1\,$ $\scriptstyle {\frac {x}{(1-x)}}\sum _{i=0}^{-1}{\binom {1}{i}}G(x^{i}),\ G(x^{0})\,=\,G(1)\,=\,{\frac {1}{1-x}}\,$	$\infty \,$	$0,\ n\neq 1,\,$ $1,\ n=1.\,$	$H_{m}^{(0)}\,$ $m\,$	$H_{m}^{(0)}\,$ $m\,$	$\scriptstyle \lim _{m\to \infty }H_{m}^{(0)}\,$ $\scriptstyle \,\sim \,m\,\to \,\infty \,$
1	Segment numbers 1-orthotope di-0-cell (2) {}	${\frac {x\ A_{1}(x)}{(1-x)^{2}}},\,$ with $\scriptstyle A_{1}(x)\,=\,1\,$ $\scriptstyle {\frac {x}{(1-x)}}\sum _{i=0}^{0}{\binom {1}{i}}G(x^{i}),\ G(x^{0})\,=\,G(1)\,=\,{\frac {1}{1-x}}\,$	$1\,$	$1\,$	$P_{3}^{(2)}(m)=t_{m}$ $\scriptstyle {\binom {m+1}{2}}\,=\,{{m^{(2)}} \over 2!}\,=\,\left(\!\!{\binom {m}{2}}\!\!\right)\,$  $H_{m}^{(-1)}\,$	$H_{m}=H_{m}^{(1)}\,$	$\scriptstyle \lim _{m\to \infty }H_{m}\,$ $\scriptstyle \,\sim \,log(m)\,\to \,\infty \,$
2	Square numbers 2-orthotope tetra-1-cell (4, 4) {4}	${\frac {x\ A_{2}(x)}{(1-x)^{3}}},\,$ with $\scriptstyle A_{2}(x)\,=\,1+x\,$ $\scriptstyle {\frac {x}{(1-x)}}\sum _{i=0}^{1}{\binom {2}{i}}G(x^{i}),\ G(x^{0})\,=\,G(1)\,=\,{\frac {1}{1-x}}\,$	$4\,$	$2n-1\,$ Odd numbers A005408	${1 \over 4}{\binom {2m+2}{3}}\,$ ${\frac {(2m)^{(3)}}{4!}}\,$ $H_{m}^{(-2)}\,$	$H_{m}^{(2)}\,$	$\zeta (2)={{\pi ^{2}} \over 6}\,$ Base 10: A013661 CFrac: A013679
3	Cube numbers 3-orthotope hexa-2-cell (8, 12, 6) {4, 3}	${\frac {x\ A_{3}(x)}{(1-x)^{4}}}\,$ with $\scriptstyle A_{3}(x)=1+4x+x^{2}\,$ $\scriptstyle {\frac {x}{(1-x)}}\sum _{i=0}^{2}{\binom {3}{i}}G(x^{i}),\ G(x^{0})\,=\,G(1)\,=\,{\frac {1}{1-x}}\,$	$9\,$	$3(n-1)^{2}+3(n-1)+1\,$ Hex (or centered hexagonal) numbers A003215	$T_{m}^{2}\,$ $H_{m}^{(-3)}\,$	$H_{m}^{(3)}\,$	$\zeta (3)\,$ Base 10: A002117 CFrac: A013631
4	Tesseract numbers 4-orthotope octa-3-cell (16, 32, 24, 8) {4, 3, 3}	${\frac {x\ A_{4}(x)}{(1-x)^{5}}}\,$ with $\scriptstyle A_{4}(x)\,=\,1+11x+11x^{2}+x^{3}\,$ or $\scriptstyle A_{4}(x)\,=\,(1+x)(1+10x+x^{2})\,$ $\scriptstyle {\frac {x}{(1-x)}}\sum _{i=0}^{3}{\binom {4}{i}}G(x^{i}),\ G(x^{0})\,=\,G(1)\,=\,{\frac {1}{1-x}}\,$	$19\,$	$\scriptstyle 4(n-1)^{3}+6(n-1)^{2}+4(n-1)+1\,$ Rhombic dodecahedral numbers A005917	$\scriptstyle {{(2m)^{(3)}} \over 5!}(6T_{m}-1)\,$ $H_{m}^{(-4)}\,$	$H_{m}^{(4)}\,$	$\zeta (4)={{\pi ^{4}} \over 90}\,$ Base 10: A013662 CFrac: A013680
5	Penteract numbers 5-orthotope deca-4-cell (32, 80, 80, 40, 10) {4, 3, 3, 3}	${\frac {x\ A_{5}(x)}{(1-x)^{6}}}\,$ with $\scriptstyle A_{5}(x)\,=\,1+26x+66x^{2}+26x^{3}+x^{4}\,$ $\scriptstyle {\frac {x}{(1-x)}}\sum _{i=0}^{4}{\binom {5}{i}}G(x^{i}),\ G(x^{0})\,=\,G(1)\,=\,{\frac {1}{1-x}}\,$	$37\,$	$\scriptstyle 5(n-1)^{4}+10(n-1)^{3}+10(n-1)^{2}+5(n-1)+1\,$	$\scriptstyle {{T_{m}^{2}} \over 3}(2m^{2}+2m-1)\,$ $H_{m}^{(-5)}\,$	$H_{m}^{(5)}\,$	$\zeta (5)\,$ Base 10: A013663 CFrac: A013681
6	Hexeract numbers 6-orthotope dodeca-5-cell (64, 192, 240, 160, 60, 12) {4, 3, 3, 3, 3}	${\frac {x\ A_{6}(x)}{(1-x)^{7}}}\,$ with $\scriptstyle A_{6}(x)\,=\,1+57x+302x^{2}+302x^{3}+57x^{4}+x^{5}\,$ or $\scriptstyle A_{6}(x)\,=\,(1+x)(1+56x+246x^{2}+56x^{3}+x^{4})\,$ $\scriptstyle {\frac {x}{(1-x)}}\sum _{i=0}^{5}{\binom {6}{i}}G(x^{i}),\ G(x^{0})\,=\,G(1)\,=\,{\frac {1}{1-x}}\,$	$73\,$	$\sum _{i=1}^{6}{\binom {6}{i}}(n-1)^{6-i}\,$	$H_{m}^{(-6)}\,$	$H_{m}^{(6)}\,$	$\zeta (6)={{\pi ^{6}} \over 945}\,$ Base 10: A013664 CFrac: A013682
7	Hepteract numbers 7-orthotope tetradeca-6-cell (128, 448, 672, 560, 280 , 84, 14) {4, 3, 3, 3, 3, 3}	${\frac {x\ A_{7}(x)}{(1-x)^{8}}},\,$ with $\scriptstyle A_{7}(x)\,=\,\,$ $\scriptstyle 1+120x+1191x^{2}+2416x^{3}+1191x^{4}+120x^{5}+x^{6}\,$ $\scriptstyle {\frac {x}{(1-x)}}\sum _{i=0}^{6}{\binom {7}{i}}G(x^{i}),\ G(x^{0})\,=\,G(1)\,=\,{\frac {1}{1-x}}\,$	$143\,$	$\sum _{i=1}^{7}{\binom {7}{i}}(n-1)^{7-i}\,$	$H_{m}^{(-7)}\,$	$H_{m}^{(7)}\,$	$\zeta (7)\,$ Base 10: A013665 CFrac: A013683
8	Octeract numbers 8-orthotope hexadeca-7-cell (256, 1024 , 1792, 1792, 1120, 448, 112, 16) {4, 3, 3, 3, 3, 3, 3}	${\frac {x\ A_{8}(x)}{(1-x)^{9}}},\,$ with $\scriptstyle A_{8}(x)\,=\,\sum _{k=1}^{8}A(8,k)\ x^{k-1},\,$ where $\scriptstyle A(8,k)|_{k=1}^{8}\,=\,\,$ $\scriptstyle \{1,247,4293,15619,15619,4293,247,1\}\,$ $\scriptstyle {\frac {x}{(1-x)}}\sum _{i=0}^{7}{\binom {8}{i}}G(x^{i}),\ G(x^{0})\,=\,G(1)\,=\,{\frac {1}{1-x}}\,$	$279\,$	$\sum _{i=1}^{8}{\binom {8}{i}}(n-1)^{8-i}\,$	$H_{m}^{(-8)}\,$	$H_{m}^{(8)}\,$	$\zeta (8)={{\pi ^{8}} \over 9450}\,$ Base 10: A013666 CFrac: A013684
9	Enneract numbers 9-orthotope octadeca-8-cell (..., 18) {4, 3, 3, 3, 3, 3, 3, 3}	${\frac {x\ A_{9}(x)}{(1-x)^{10}}}\,$ with $\scriptstyle A_{9}(x)\,=\,\sum _{k=1}^{9}A(9,k)\ x^{k-1},\,$ where $\scriptstyle A(9,k)|_{k=1}^{9}\,=\,\,$ $\scriptstyle \{1,502,14608,88234,156190,88234,14608,502,1\}\,$ $\scriptstyle {\frac {x}{(1-x)}}\sum _{i=0}^{8}{\binom {9}{i}}G(x^{i}),\ G(x^{0})\,=\,G(1)\,=\,{\frac {1}{1-x}}\,$	$548\,$	$\sum _{i=1}^{9}{\binom {9}{i}}(n-1)^{9-i}\,$	$H_{m}^{(-9)}\,$	$H_{m}^{(9)}\,$	$\zeta (9)\,$ Base 10: A013667 CFrac: A013685
10	Dekeract numbers 10-orthotope icosa-9-cell (..., 20) {4, 3, 3, 3, 3, 3, 3, 3, 3}	${\frac {x\ A_{10}(x)}{(1-x)^{11}}}\,$ with $\scriptstyle A_{10}(x)\,=\,\sum _{k=1}^{10}A(10,k)\ x^{k-1},\,$ where $\scriptstyle A(10,k)|_{k=1}^{10}\,=\,\,$ $\scriptstyle \{1,1013,47840,455192,1310354,\,$ $\scriptstyle 1310354,455192,47840,1013,1\}\,$ $\scriptstyle {\frac {x}{(1-x)}}\sum _{i=0}^{9}{\binom {10}{i}}G(x^{i}),\ G(x^{0})\,=\,G(1)\,=\,{\frac {1}{1-x}}\,$	$1079\,$	$\sum _{i=1}^{10}{\binom {10}{i}}(n-1)^{10-i}\,$	$H_{m}^{(-10)}\,$	$H_{m}^{(10)}\,$	$\zeta (10)=\,$ ${{\pi ^{10}} \over 93555}\,$ Base 10: A013668 CFrac: A013686
11	Hendekeract numbers 11-orthotope icosidi-10-cell (..., 22) {4, 3, 3, 3, 3, 3, 3, 3, 3, 3}	${\frac {x\ A_{11}(x)}{(1-x)^{12}}}\,$ with $\scriptstyle A_{11}(x)\,=\,\sum _{k=1}^{11}A(11,k)\ x^{k-1},\,$ where $\scriptstyle A(11,k)|_{k=1}^{11}\,=\,\,$ $\scriptstyle \{1,2036,...,2036,1\}\,$ $\scriptstyle {\frac {x}{(1-x)}}\sum _{i=0}^{10}{\binom {11}{i}}G(x^{i}),\ G(x^{0})\,=\,G(1)\,=\,{\frac {1}{1-x}}\,$	$2132\,$	$\sum _{i=1}^{11}{\binom {11}{i}}(n-1)^{11-i}\,$	$H_{m}^{(-11)}\,$	$H_{m}^{(11)}\,$	$\zeta (11)\,$ Base 10: A013669 CFrac: A013687
12	Dodekeract numbers 12-orthotope icositetra-11-cell (..., 24) {4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3}	${\frac {x\ A_{12}(x)}{(1-x)^{13}}}\,$ with $\scriptstyle A_{12}(x)\,=\,\sum _{k=1}^{12}A(12,k)\ x^{k-1},\,$ where $\scriptstyle A(12,k)|_{k=1}^{12}\,=\,\,$ $\scriptstyle \{1,4083,...,4083,1\}\,$ $\scriptstyle {\frac {x}{(1-x)}}\sum _{i=0}^{11}{\binom {12}{i}}G(x^{i}),\ G(x^{0})\,=\,G(1)\,=\,{\frac {1}{1-x}}\,$	$4223\,$	$\sum _{i=1}^{12}{\binom {12}{i}}(n-1)^{12-i}\,$	$H_{m}^{(-12)}\,$	$H_{m}^{(12)}\,$	$\zeta (12)=\,$ ${{691\pi ^{12}} \over 638512875}\,$ Base 10: A013670 CFrac: A013688

Table of sequences

Regular orthotopic numbers sequences

d	A-number	P  (d  ) 2 d(n), n   ≥   0
0	(NOT A057427)	{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, ...}
1	A001477	{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	A000290	{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	A000578	{0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 13824, 15625, 17576, 19683, 21952, 24389, ...}
4	A000583	{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, ...}
5	A000584	{0, 1, 32, 243, 1024, 3125, 7776, 16807, 32768, 59049, 100000, 161051, 248832, 371293, 537824, 759375, 1048576, 1419857, 1889568, 2476099, 3200000, 4084101, 5153632, ...}
6	A001014	{0, 1, 64, 729, 4096, 15625, 46656, 117649, 262144, 531441, 1000000, 1771561, 2985984, 4826809, 7529536, 11390625, 16777216, 24137569, 34012224, 47045881, 64000000, ...}
7	A001015	{0, 1, 128, 2187, 16384, 78125, 279936, 823543, 2097152, 4782969, 10000000, 19487171, 35831808, 62748517, 105413504, 170859375, 268435456, 410338673, 612220032, ...}
8	A001016	{0, 1, 256, 6561, 65536, 390625, 1679616, 5764801, 16777216, 43046721, 100000000, 214358881, 429981696, 815730721, 1475789056, 2562890625, 4294967296, 6975757441, ...}
9	A001017	{0, 1, 512, 19683, 262144, 1953125, 10077696, 40353607, 134217728, 387420489, 1000000000, 2357947691, 5159780352, 10604499373, 20661046784, 38443359375, ...}
10	A008454	{0, 1, 1024, 59049, 1048576, 9765625, 60466176, 282475249, 1073741824, 3486784401, 10000000000, 25937424601, 61917364224, 137858491849, 289254654976, ...}
11	A008455	{0, 1, 2048, 177147, 4194304, 48828125, 362797056, 1977326743, 8589934592, 31381059609, 100000000000, 285311670611, 743008370688, 1792160394037, 4049565169664, ...}
12	A008456	{0, 1, 4096, 531441, 16777216, 244140625, 2176782336, 13841287201, 68719476736, 282429536481, 1000000000000, 3138428376721, 8916100448256, 23298085122481, ...}

Regular orthotopic numbers read cross-dimensionally (giving exponentials sequences)

The regular orthotopic numbers read cross-dimensionally give the exponentials sequences.

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).

Regular orthotopic numbers read cross-dimensionally (giving exponentials sequences)

b	A-number	P  (n) 2 n(b) = b n, n   ≥   0
0	A000007	{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	{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	A000079	{1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, ...}
3	A000244	{1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907, 43046721, 129140163, 387420489, 1162261467, ...}
4	A000302	{1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144, 1048576, 4194304, 16777216, 67108864, 268435456, 1073741824, 4294967296, 17179869184, ...}
5	A000351	{1, 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, 9765625, 48828125, 244140625, 1220703125, 6103515625, 30517578125, 152587890625, ...}
6	A000400	{1, 6, 36, 216, 1296, 7776, 46656, 279936, 1679616, 10077696, 60466176, 362797056, 2176782336, 13060694016, 78364164096, 470184984576, ...}
7	A000420	{1, 7, 49, 343, 2401, 16807, 117649, 823543, 5764801, 40353607, 282475249, 1977326743, 13841287201, 96889010407, 678223072849, 4747561509943, ...}
8	A001018	{1, 8, 64, 512, 4096, 32768, 262144, 2097152, 16777216, 134217728, 1073741824, 8589934592, 68719476736, 549755813888, 4398046511104, ...}
9	A001019	{1, 9, 81, 729, 6561, 59049, 531441, 4782969, 43046721, 387420489, 3486784401, 31381059609, 282429536481, 2541865828329, 22876792454961, ...}
10	A011557	{1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000, 10000000000, 100000000000, 1000000000000, 10000000000000, ...}
11	A001020	{1, 11, 121, 1331, 14641, 161051, 1771561, 19487171, 214358881, 2357947691, 25937424601, 285311670611, 3138428376721, 34522712143931, ...}
12	A001021	{1, 12, 144, 1728, 20736, 248832, 2985984, 35831808, 429981696, 5159780352, 61917364224, 743008370688, 8916100448256, 106993205379072, ...}

See also

• Centered regular orthotopic numbers

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.