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# Centered regular orthotopic numbers

The centered regular orthotopic numbers are a family of sequences of...

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

## Formulae

The nth d-dimensional centered regular orthotopic number is given by the formula:

$\,_cP^{(d)}_{2^d}(n) = P^{(d)}_{2^d}(n) + P^{(d)}_{2^d}(n+1) = n^d + (n+1)^d,\,$

where d is the dimension and 2d is the number of vertices.

## Schläfli-Poincaré (convex) polytope formula

Generalization for polytopes of Descartes-Euler (convex) polyhedral formula:[1]

${\sum_{i=0}^{d-1} (-1)^i N_i} = 1 - (-1)^d,\,$

where N0 is the number of 0-dimensional elements, N1 is the number of 1-dimensional elements, N2 is the number of 2-dimensional elements...

## Recurrence equation

$\,_cP^{(d)}_{2^d}(n) = (n+1) \,_cP^{(d-1)}_{2^{d-1}}(n) - P^{(d-1)}_{2^{d-1}}(n) = (n+1) \,_cP^{(d-1)}_{2^{d-1}}(n) - n^{d-1},\,$

with initial conditions

$\,_cP^{(1)}_{2^1}(n) = P^{(1)}_{2}(n) = 2n+1.\,$

## Generating function

$G_{\{\,_cP^{(d)}_{2^d}(n)\}}(x) = G_{\{n^d + (n+1)^d\}}(x) = G_{\{n^d\}}(x) + G_{\{(n+1)^d\}}(x) = G_{\{n^d\}}(x) + \frac{G_{\{n^d\}}(x)}{x} = \frac{(x+1)}{x} G_{\{n^d\}}(x) = \frac{(1+x)}{x} G_{\{P^{(d)}_{2^d}\}}(x)\,$

From the generating function of regular orthotopic numbers, we obtain:

$G_{\{\,_cP^{(d)}_{2^d}(n)\}}(x) = \frac{(1+x)}{x} \bigg\{\frac{x}{(1-x)} \sum_{i=0}^{d-1} \binom{d}{i} G_{\{n^i\}}(x)\bigg\} = \frac{(1+x)}{(1-x)} \sum_{i=0}^{d-1} \binom{d}{i} G_{\{n^i\}}(x),\,$ with $G_{\{n^0\}}(x) = G_{\{1\}}(x) = \frac{1}{1-x}\,$
$G_{\{\,_cP^{(d)}_{2^d}(n)\}}(x) = \frac{(1+x)A_{d}(x)}{(1-x)^{d+1}} = \frac{(1+x)[1+(2^d-d-1)x+...+(2^d-d-1)x^{d-2}+x^{d-1}]}{(1-x)^{d+1}},\,$

where $\scriptstyle A_{d}(x)\,$ is the dth, d ≥ 0, Eulerian polynomial.

## 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.[2] 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 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 $\scriptstyle k \ge 3$, the set $\scriptstyle \{P(k, n)|n = 0, 1, 2, \ldots\}\,$ of k-gon numbers forms a basis of order k, i.e. every nonnegative integer can be written as a sum of 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.[3] 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 $\scriptstyle g(d)\,$ for $\scriptstyle d \ge 2\,$. 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

$\,_cP^{(d)}_{2^d}(n) - \,_cP^{(d)}_{2^d}(n-1) = P^{(d)}_{2^d}(n+1) - P^{(d)}_{2^d}(n-1) = (n+1)^d - (n-1)^d\,$

## Partial sums

$\sum_{n=1}^m \,_cP^{(d)}_{2^d}(n) = ?,\,$

where $\scriptstyle t_m\,$ is the mth triangular number.

## Partial sums of reciprocals

$\sum_{n=1}^m \frac{1}{\,_cP^{(d)}_{2^d}(n)} = ?\,$

## Sum of reciprocals

$\sum_{n=1}^\infty \frac{1}{\,_cP^{(d)}_{2^d}(n)} = ?\,$

## 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-2)-dimensional elements are the polytope ridges and the (d-1)-dimensional elements are the polytope facets. The centered regular orthotopic numbers are listed by increasing number N0 of vertices.

Centered regular orthotopic numbers formulae and values
d Name

Regular

d-orthotope

2d (d-1)-cell

(N0, N1, N2, ...)

Schläfli symbol[4]

Formulae

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

$\scriptstyle n^d+(n+1)^d\,$

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

number

1 Centered 2-step gnomonic numbers

1-orthotope

di-0-cell

(2)

{}

$\scriptstyle n+(n+1)\,$

$\scriptstyle 2n+1\,$

1 3 5 7 9 11 13 15 17 19 21 23 25 A005408(n)
2 Centered square numbers

2-orthotope

tetra-1-cell

(4, 4)

{4}

$\scriptstyle n^2+(n+1)^2\,$

$\scriptstyle 2n(n+1)+1\,$

$\scriptstyle 4 T_n+1\,$

1 5 13 25 41 61 85 113 145 181 221 265 313 A001844(n)
3 Centered cube numbers

3-orthotope

hexa-2-cell

(8, 12, 6)

{4, 3}

$\scriptstyle n^3 + (n+1)^3\,$ 1 9 35 91 189 341 559 855 1241 1729 2331 3059 3925 A005898(n)
4 Centered tesseract numbers

4-orthotope

octa-3-cell

(16, 32, 24, 8)

{4, 3, 3}

$\scriptstyle n^4 + (n+1)^4\,$ 1 17 97 337 881 1921 3697 6497 10657 16561 24641 35377 49297 A008514(n)
5 Centered penteract numbers

5-orthotope

deca-4-cell

(32, 80, 80, 40, 10)

{4, 3, 3, 3}

$\scriptstyle n^5 + (n+1)^5\,$ 1 33 275 1267 4149 10901 24583 49575 91817 159049 261051 409883 620125 A008515(n)
6 Centered hexeract numbers

6-orthotope

dodeca-5-cell

(64, 192, 240, 160, 60, 12)

{4, 3, 3, 3, 3}

$\scriptstyle n^6 + (n+1)^6\,$ 1 65 793 4825 19721 62281 164305 379793 793585 1531441 2771561 4757545 7812793 A008516(n)
7 Centered hepteract numbers

7-orthotope

(128, 448, 672, 560, 280, 84, 14)

{4, 3, 3, 3, 3, 3}

$\scriptstyle n^7 + (n+1)^7\,$ 1 129 2315 18571 94509 358061 1103479 2920695 6880121 14782969 29487171 55318979 98580325 A036085(n)
8 Centered octeract numbers

8-orthotope

(256, 1024, 1792, 1792, 1120, 448, 112, 16)

{4, 3, 3, 3, 3, 3, 3}

$\scriptstyle n^8 + (n+1)^8\,$ 1 257 6817 72097 456161 2070241 7444417 22542017 59823937 143046721 314358881 644340577 1245712417 A036086(n)
9 Centered enneract numbers

9-orthotope

(512, 2304, 4608, 5376, 4032, 2016, 672, 144, 18)

{4, 3, 3, 3, 3, 3, 3, 3}

$\scriptstyle n^9 + (n+1)^9\,$ 1 513 20195 281827 2215269 12030821 50431303 174571335 521638217 1387420489 3357947691 7517728043 15764279725 A036087(n)
10 Centered 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}

$\scriptstyle n^{10} + (n+1)^{10}\,$ 1 1025 60073 1107625 10814201 70231801 342941425 1356217073 4560526225 13486784401 35937424601 87854788825 199775856073 A036088(n)
11 Centered hendekeract numbers

11-orthotope

icosidi-10-cell

(2048, ..., 22)

{4, 3, 3, 3, 3, 3, 3, 3, 3, 3}

$\scriptstyle n^{11} + (n+1)^{11}\,$ 1 2049 179195 4371451 53022429 411625181 2340123799 10567261335 39970994201 131381059609 385311670611 1028320041299 2535168764725 A036089(n)
12 Centered dodekeract numbers

12-orthotope

icositetra-11-cell

(4096, ..., 24)

{4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3}

$\scriptstyle n^{12} + (n+1)^{12}\,$ 1 4097 535537 17308657 260917841 2420922961 16018069537 82560763937 351149013217 1282429536481 4138428376721 12054528824977 32214185570737 A036090(n)

## Table of related formulae and values

N0, N1, N2,N3, ... are the number of vertices (0-dimensional), edges (1-dimensional), faces (2-dimensional), cells (3-dimensional)... respectively, where the (n-1)-dimensional "vertices" are the actual facets. The centered regular orthotopic numbers are listed by increasing number N0 of vertices.

Centered regular orthotopic numbers related formulae and values
d Name

Regular d-orthotope

2d (d-1)-cell

(N0, N1, N2, ...)

Schläfli symbol[4]

Generating

function

$G_{\{\,_cP^{(d)}_{2^d}(n)\}}(x) = \,$

$\scriptstyle {\big(\frac{1+x}{x}\big)\,G_{\{P^{(d)}_{2^d}\}}(x)\,=\,\big(\frac{1+x}{1-x}\big) \sum_{i=0}^{d-1} \binom{d}{i} G(x^i)}\,$ [5]

$\scriptstyle {\frac{(1+x)A_{d}(x)}{(1-x)^{d+1}}}\,$

$\scriptstyle {{(1+x)(1+(2^d-d-1)x+...+(2^d-d-1)x^{d-2}+x^{d-1})}\over{(1-x)^{d+1}}}\,$

where $\scriptstyle A_{d}(x)\,$ is the dth, d ≥ 0, Eulerian polynomial.

Order

of basis

$g_{\{\,_cP^{(d)}_{2^d}\}}\,$

Differences

$\,_cP^{(d)}_{2^d}(n) - \,$

$\,_cP^{(d)}_{2^d}(n-1) =\,$

$\scriptstyle (n+1)^d - (n-1)^d\,$

Partial sums

$\sum_{n=1}^m {\,_cP^{(d)}_{2^d}(n)} =\,$

Partial sums of reciprocals

$\sum_{n=1}^m {1\over{\,_cP^{(d)}_{2^d}(n)}} =\,$

Sum of reciprocals[6]

$\sum_{n=1}^\infty {1\over{P^{(d)}_{2^d}(n)}} = \,$

1 Centered 2-step gnomonic numbers

1-orthotope

di-0-cell

(2)

{}

$\scriptstyle \frac{(1+x)}{(1-x)} \sum_{i=0}^{0} \binom{1}{i} G(x^i)\,$

$\scriptstyle {\frac{(1+x)A_{1}(x)}{(1-x)^2}},\,$

with $\scriptstyle A_{1}(x) = 1\,$

$\,$ $\scriptstyle (n+1) - (n-1)\,$

$2\,$

$\,$ $\,$ $\,$
2 Centered square numbers

2-orthotope

tetra-1-cell

(4, 4)

{4}

$\scriptstyle \frac{(1+x)}{(1-x)} \sum_{i=0}^{1} \binom{2}{i} G(x^i)\,$

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

with $\scriptstyle A_{2}(x) = 1+x\,$

$\,$ $\scriptstyle (n+1)^2 - (n-1)^2\,$ $\,$ $\,$ $\,$
3 Centered cube numbers

3-orthotope

hexa-2-cell

(8, 12, 6)

{4, 3}

$\scriptstyle \frac{(1+x)}{(1-x)} \sum_{i=0}^{2} \binom{3}{i} G(x^i)\,$

$\scriptstyle {\frac{(1+x)A_{3}(x)}{(1-x)^4}},\,$

with $\scriptstyle A_{3}(x) = 1+4x+x^2\,$

$\,$ $\scriptstyle (n+1)^3 - (n-1)^3\,$ $\,$ $\,$ $\,$
4 Centered tesseract numbers

4-orthotope

octa-3-cell

(16, 32, 24, 8)

{4, 3, 3}

$\scriptstyle \frac{(1+x)}{(1-x)} \sum_{i=0}^{3} \binom{4}{i} G(x^i)\,$

$\scriptstyle {\frac{(1+x)A_{4}(x)}{(1-x)^5}},\,$

with $\scriptstyle A_{4}(x) = 1+11x+11x^2+x^3\,$

$\,$ $\scriptstyle (n+1)^4 - (n-1)^4\,$ $\,$ $\,$ $\,$
5 Centered penteract numbers

5-orthotope

deca-4-cell

(32, 80, 80, 40, 10)

{4, 3, 3, 3}

$\scriptstyle \frac{(1+x)}{(1-x)} \sum_{i=0}^{4} \binom{5}{i} G(x^i)\,$

$\scriptstyle {\frac{(1+x)A_{5}(x)}{(1-x)^6}},\,$

with $\scriptstyle A_{5}(x) = 1+26x+66x^2+26x^3+x^4\,$

$\,$ $\scriptstyle (n+1)^5 - (n-1)^5\,$ $\,$ $\,$ $\,$
6 Centered hexeract numbers

6-orthotope

dodeca-5-cell

(64, 192, 240, 160, 60, 12)

{4, 3, 3, 3, 3}

$\scriptstyle \frac{(1+x)}{(1-x)} \sum_{i=0}^{5} \binom{6}{i} G(x^i)\,$

$\scriptstyle {\frac{(1+x)A_{6}(x)}{(1-x)^7}},\,$

with $\scriptstyle A_{6}(x) = 1+57x+302x^2+302x^3+57x^4+x^5\,$

$\,$ $\scriptstyle (n+1)^6 - (n-1)^6\,$ $\,$ $\,$ $\,$
7 Centered hepteract numbers

7-orthotope

(128, 448, 672, 560, 280 , 84, 14)

{4, 3, 3, 3, 3, 3}

$\scriptstyle \frac{(1+x)}{(1-x)} \sum_{i=0}^{6} \binom{7}{i} G(x^i)\,$

$\scriptstyle {\frac{(1+x)A_{7}(x)}{(1-x)^8}},\,$

with $\scriptstyle A_{7}(x) = 1+120x+1191x^2+2416x^3+1191x^4+120x^5+x^6\,$

$\,$ $\scriptstyle (n+1)^7 - (n-1)^7\,$ $\,$ $\,$ $\,$
8 Centered octeract numbers

8-orthotope

(256, 1024 , 1792, 1792, 1120, 448, 112, 16)

{4, 3, 3, 3, 3, 3, 3}

$\scriptstyle \frac{(1+x)}{(1-x)} \sum_{i=0}^{7} \binom{8}{i} G(x^i)\,$

$\scriptstyle {\frac{(1+x)A_{8}(x)}{(1-x)^9}},\,$

with $\scriptstyle A_{8}(x) = 1+247x+ ... +247x^6+x^7\,$

$\,$ $\scriptstyle (n+1)^8 - (n-1)^8\,$ $\,$ $\,$ $\,$
9 Centered enneract numbers

9-orthotope

(512, 2304, 4608, 5376, 4032, 2016, 672, 144, 18)

{4, 3, 3, 3, 3, 3, 3, 3}

$\scriptstyle \frac{(1+x)}{(1-x)} \sum_{i=0}^{8} \binom{9}{i} G(x^i)\,$

$\scriptstyle {\frac{(1+x)A_{9}(x)}{(1-x)^{10}}},\,$

with $\scriptstyle A_{9}(x) = 1+502x+ ... +502x^7+x^8\,$

$\,$ $\scriptstyle (n+1)^9 - (n-1)^9\,$ $\,$ $\,$ $\,$
10 Centered 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}

$\scriptstyle \frac{(1+x)}{(1-x)} \sum_{i=0}^{9} \binom{10}{i} G(x^i)\,$

$\scriptstyle {\frac{(1+x)A_{10}(x)}{(1-x)^{11}}},\,$

with $\scriptstyle A_{10}(x) = 1+1013x+ ... +1013x^8+x^9\,$

$\,$ $\scriptstyle (n+1)^{10} - (n-1)^{10}\,$ $\,$ $\,$ $\,$
11 Centered hendekeract numbers

11-orthotope

icosidi-10-cell

(2048, ..., 22)

{4, 3, 3, 3, 3, 3, 3, 3, 3, 3}

$\scriptstyle \frac{(1+x)}{(1-x)} \sum_{i=0}^{10} \binom{11}{i} G(x^i)\,$

$\scriptstyle {\frac{(1+x)A_{11}(x)}{(1-x)^{12}}},\,$

with $\scriptstyle A_{11}(x) = 1+2036x+ ... +2036x^9+x^{10}\,$

$\,$ $\scriptstyle (n+1)^{11} - (n-1)^{11}\,$ $\,$ $\,$ $\,$
12 Centered dodekeract numbers

12-orthotope

icositetra-11-cell

(4096, ..., 24)

{4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3}

$\scriptstyle \frac{(1+x)}{(1-x)} \sum_{i=0}^{11} \binom{12}{i} G(x^i)\,$

$\scriptstyle {\frac{(1+x)A_{12}(x)}{(1-x)^{13}}}\,$

with $\scriptstyle A_{12}(x) = 1+4083x+ ... +4083x^{10}+x^{11}\,$

$\,$ $\scriptstyle (n+1)^{12} - (n-1)^{12}\,$ $\,$ $\,$ $\,$

## Table of sequences

Centered regular orthotopic numbers sequences
d $\,_cP^{(d)}_{2d}(n),\ n \ge 0\,$ sequences
1 {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, ...}
2 {1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 265, 313, 365, 421, 481, 545, 613, 685, 761, 841, 925, 1013, 1105, 1201, 1301, 1405, 1513, 1625, 1741, 1861, 1985, ...}
3 {1, 9, 35, 91, 189, 341, 559, 855, 1241, 1729, 2331, 3059, 3925, 4941, 6119, 7471, 9009, 10745, 12691, 14859, 17261, 19909, 22815, 25991, 29449, 33201, 37259, 41635, ...}
4 {1, 17, 97, 337, 881, 1921, 3697, 6497, 10657, 16561, 24641, 35377, 49297, 66977, 89041, 116161, 149057, 188497, 235297, 290321, 354481, 428737, 514097, 611617, ... }
5 {1, 33, 275, 1267, 4149, 10901, 24583, 49575, 91817, 159049, 261051, 409883, 620125, 909117, 1297199, 1807951, 2468433, 3309425, 4365667, 5676099, 7284101, ...}
6 {1, 65, 793, 4825, 19721, 62281, 164305, 379793, 793585, 1531441, 2771561, 4757545, 7812793, 12356345, 18920161, 28167841, 40914785, 58149793, 81058105, ...}
7 {1, 129, 2315, 18571, 94509, 358061, 1103479, 2920695, 6880121, 14782969, 29487171, 55318979, 98580325, 168162021, 276272879, 439294831, 678774129, ...}
8 {1, 257, 6817, 72097, 456161, 2070241, 7444417, 22542017, 59823937, 143046721, 314358881, 644340577, 1245712417, 2291519777, 4038679681, 6857857921, ...}
9 {1, 513, 20195, 281827, 2215269, 12030821, 50431303, 174571335, 521638217, 1387420489, 3357947691, 7517728043, 15764279725, 31265546157, 59104406159, ...}
10 {1, 1025, 60073, 1107625, 10814201, 70231801, 342941425, 1356217073, 4560526225, 13486784401, 35937424601, 87854788825, 199775856073, 427113146825, ...}
11 {1, 2049, 179195, 4371451, 53022429, 411625181, 2340123799, 10567261335, 39970994201, 131381059609, 385311670611, 1028320041299, 2535168764725, ...}
12 {1, 4097, 535537, 17308657, 260917841, 2420922961, 16018069537, 82560763937, 351149013217, 1282429536481, 4138428376721, 12054528824977, 32214185570737, ...}

## Notes

1. Weisstein, Eric W., Polyhedral Formula, From MathWorld--A Wolfram Web Resource.
2. Weisstein, Eric W., Fermat's Polygonal Number Theorem, From MathWorld--A Wolfram Web Resource.
3. 3.0 3.1 Weisstein, Eric W., Waring's Problem, From MathWorld--A Wolfram Web Resource.
4. 4.0 4.1 Weisstein, Eric W., Schläfli Symbol, From MathWorld--A Wolfram Web Resource.
5. Generating function of regular orthotopic numbers.
6. Downey, Lawrence M., Ong, Boon W., and Sellers, James A., Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, 2008.