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Centered orthoplex numbers

(Redirected from Centered cross polytope numbers)

The centered orthoplicial polytopic numbers are a family of sequences of...

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Formulae

The nth d-dimensional centered orthoplicial polytopic number is given by the formula:

${\displaystyle \,_{c}P^{(d)}(2d,n)=?,\,}$

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]

${\displaystyle {\sum _{i=0}^{d-1}(-1)^{i}N_{i}}=1-(-1)^{d},\,}$

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

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

with initial conditions:

${\displaystyle \,_{c}P^{(d)}(2d,0)=1,\,}$
${\displaystyle \,_{c}P^{(1)}(2\cdot 1,n)=2n+1.\,}$

Generating function

${\displaystyle G_{\,_{c}P^{(d)}}(2d,x)={\frac {(1+x)^{d}}{(1-x)^{d+1}}}\,}$

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-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 ${\displaystyle \scriptstyle \{n^{2}|n=0,1,2,\ldots \}\,}$ 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 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. 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

${\displaystyle \,_{c}P^{(d)}(2d,n)-\,_{c}P^{(d)}(2d,n-1)=?\,}$

Partial sums

${\displaystyle \sum _{n=0}^{m}\,_{c}P^{(d)}(2d,n)=?,\,}$

where ${\displaystyle \scriptstyle t_{m}\,}$ is the mth triangular number.

Partial sums of reciprocals

${\displaystyle \sum _{n=0}^{m}{\frac {1}{\,_{c}P^{(d)}(2d,n)}}=?\,}$

Sum of reciprocals

${\displaystyle \sum _{n=0}^{\infty }{\frac {1}{\,_{c}P^{(d)}(2d,n)}}=?\,}$

Table of 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 orthoplicial numbers are listed by increasing number N0 of vertices.

Centered orthoplicial numbers formulae and values
d Name

Regular

d-orthoplex

2d (d-1)-cell

(N0, N1, N2, ...)

Schläfli symbol[3]

Formulae

${\displaystyle \,_{c}P_{2d}^{(d)}(n)=?\,}$

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

number

1 Centered square gnomon

1-orthoplex

di-0-cell

(2)

{}

${\displaystyle n+(n+1)\,}$

${\displaystyle 2n+1\,}$

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

2-orthoplex

Tetragon

Bicross

(4, 4)

{4}

${\displaystyle 4T_{n}+1\,}$

${\displaystyle 2n(n+1)+1\,}$

${\displaystyle n^{2}+(n+1)^{2}\,}$

${\displaystyle 1+2n+2n^{2}\,}$

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

3-orthoplex

Octahedron

Tricross

(6, 12, 8)

{3, 4}

${\displaystyle {\frac {(2n+1)(2n^{2}+2n+3)}{3}}\,}$

${\displaystyle {\frac {3+8n+6n^{2}+4n^{3}}{3}}\,}$

1 7 25 63 129 231 377 575 833 1159 1561 2047 2625 A001845
4 Centered tetracross

4-orthoplex

24 3-cell

(8, 24, 32, 16)

{3, 3, 4}

${\displaystyle {\frac {3+8n+10n^{2}+4n^{3}+2n^{4}}{3}}\,}$ 1 9 41 129 321 681 1289 2241 3649 5641 8361 11969 16641 A001846
5 Centered pentacross

5-orthoplex

25 4-cell

(10, 40, 80, 80, 32)

{3, 3, 3, 4}

${\displaystyle \scriptstyle {\frac {15+46n+50n^{2}+40n^{3}+10n^{4}+4n^{5}}{15}}\,}$

${\displaystyle \scriptstyle {\frac {(2n+1)(2n(n+1)(n^{2}+n+8)+15)}{15}}\,}$

1 11 61 231 681 1683 3653 7183 13073 22363 36365 56695 85305 A001847
6 Centered hexacross

6-orthoplex

26 5-cell

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

{3, 3, 3, 3, 4}

${\displaystyle \scriptstyle {\frac {45+138n+196n^{2}+120n^{3}+70n^{4}+12n^{5}+4n^{6}}{45}}\,}$ 1 13 85 377 1289 3653 8989 19825 40081 75517 134245 227305 369305 A001848
7 Centered heptacross

7-orthoplex

27 6-cell

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

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

${\displaystyle \scriptstyle {\frac {315+1056n+1372n^{2}+1232n^{3}+490n^{4}+224n^{5}+28n^{6}+8n^{7}}{315}}\,}$ 1 15 113 575 2241 7183 19825 48639 108545 224143 433905 795455 1392065 A001849
8 Centered octacross

8-orthoplex

28 7-cell

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

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

${\displaystyle \scriptstyle {\frac {315+1056n+1636n^{2}+1232n^{3}+798n^{4}+224n^{5}+84n^{6}+8n^{7}+2n^{8}}{315}}\,}$ 1 17 145 833 3649 13073 40081 108545 265729 598417 1256465 2485825 4673345 A008417
9 Centered enneacross

9-orthoplex

29 8-cell

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

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

${\displaystyle \scriptstyle {\frac {2835+10134n+14724n^{2}+14360n^{3}+7182n^{4}}{2835}}\,+\,}$

${\displaystyle \scriptstyle {\frac {3612n^{5}+756n^{6}+240n^{7}+18n^{8}+4n^{9}}{2835}}\,}$

1 19 181 1159 5641 22363 75517 224143 598417 1462563 3317445 7059735 14218905 A008419
10 Centered decacross

10-orthoplex

210 9-cell

(20, 180, 960, 3360, 8064, 13440, 15360, 11520, 5120, 1024)

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

${\displaystyle \scriptstyle {\frac {14175+50670n+83754n^{2}+71800n^{3}+50270n^{4}}{14175}}\,+\,}$

${\displaystyle \scriptstyle {\frac {18060n^{5}+7392n^{6}+1200n^{7}+330n^{8}+20n^{9}+4n^{10}}{14175}}\,}$

1 21 221 1561 8361 36365 134245 433905 1256465 3317445 8097453 18474633 39753273 A008421
11 Centered hendecacross

11-orthoplex

211 10-cell

(22, ..., 2048)

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

${\displaystyle \scriptstyle {\frac {155925+585720n+921294n^{2}+957308n^{3}+552970n^{4}+299200n^{5}}{155925}}\,+\,}$

${\displaystyle \scriptstyle {\frac {81312n^{6}+27984n^{7}+3630n^{8}+880n^{9}+44n^{10}+8n^{11}}{155925}}\,}$

1 23 265 2047 11969 56695 227305 795455 2485825 7059735 18474633 45046719 103274625 A240876
12 Centered dodecacross

12-orthoplex

212 11-cell

(24, ..., 4096)

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

${\displaystyle \scriptstyle {\frac {467775+1757160n+3056742n^{2}+2871924n^{3}+2137564n^{4}+897600n^{5}}{467775}}\,+\,}$

${\displaystyle \scriptstyle {\frac {393536n^{6}+83952n^{7}+24882n^{8}+2640n^{9}+572n^{10}+24n^{11}+4n^{12}}{467775}}\,}$

1 25 313 2625 16641 85305 369305 1392065 4673345 14218905 39753273 103274625 251595969 A053805

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 orthoplicial numbers are listed by increasing number N0 of vertices.

Centered orthoplicial numbers related formulae and values
d Name

Regular d-orthoplex

2d (d-1)-cell

(N0, N1, N2, ...)

Schläfli symbol[3]

Generating

function

${\displaystyle G_{\,_{c}P_{2d}^{(d)}}(x)=\,}$

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

Order

of basis

${\displaystyle g_{\,_{c}P_{2d}^{(d)}}\,}$

Differences

${\displaystyle \,_{c}P_{2d}^{(d)}(n)-\,}$

${\displaystyle \,_{c}P_{2d}^{(d)}(n-1)=\,}$

${\displaystyle ?\,}$

Partial sums

${\displaystyle \sum _{n=0}^{m}{\,_{c}P_{2d}^{(d)}(n)}=?\,}$

Partial sums of reciprocals

${\displaystyle \sum _{n=0}^{m}{1 \over {\,_{c}P_{2d}^{(d)}(n)}}=?\,}$

Sum of reciprocals[6]

${\displaystyle \sum _{n=0}^{\infty }{1 \over {\,_{c}P_{2d}^{(d)}(n)}}=?\,}$

1 Centered square gnomon

1-orthoplex

di-0-cell

(2)

{}

${\displaystyle {\frac {(1+x)}{(1-x)^{2}}}\,}$ ${\displaystyle 2\,}$ ${\displaystyle 2\,}$ ${\displaystyle (m+1)^{2}\,}$ ${\displaystyle {\frac {(\psi (m+{\tfrac {3}{2}})+\gamma )}{2}}+\log(2)\,}$ [7] [8] ${\displaystyle \infty \,}$
2 Centered square

2-orthoplex

Tetra-1-cell

(4, 4)

{4}

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

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

${\displaystyle \,}$ ${\displaystyle 4n\,}$ ${\displaystyle {\frac {(m+1)(2m^{2}+4m+3)}{3}}\,}$ ${\displaystyle \,}$ ${\displaystyle {\frac {\pi }{2}}\tanh {\bigg (}{\frac {\pi }{2}}{\bigg )}-1\,}$
3 Centered octahedral

3-orthoplex

Octa-2-cell

(6, 12, 8)

{3, 4}

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

${\displaystyle {\frac {(1+x)(1+2x+x^{2})}{(1-x)^{4}}}\,}$

${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle {\frac {(m+1)^{2}(m^{2}+2m+3)}{3}}\,}$ ${\displaystyle \,}$ ${\displaystyle \,}$
4 Centered tetracross

4-orthoplex

24 3-cell

(8, 24, 32, 16)

{3, 3, 4}

${\displaystyle {\frac {(1+x)^{4}}{(1-x)^{5}}}\,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$
5 Centered pentacross

5-orthoplex

25 4-cell

(10, 40, 80, 80, 32)

{3, 3, 3, 4}

${\displaystyle {\frac {(1+x)^{5}}{(1-x)^{6}}}\,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$
6 Centered hexacross

6-orthoplex

26 5-cell

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

{3, 3, 3, 3, 4}

${\displaystyle {\frac {(1+x)^{6}}{(1-x)^{7}}}\,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$
7 Centered heptacross

7-orthoplex

27 6-cell

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

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

${\displaystyle {\frac {(1+x)^{7}}{(1-x)^{8}}}\,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$
8 Centered octacross

8-orthoplex

28 7-cell

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

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

${\displaystyle {\frac {(1+x)^{8}}{(1-x)^{9}}}\,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$
9 Centered enneacross

9-orthoplex

29 8-cell

(18, ..., 512)

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

${\displaystyle {\frac {(1+x)^{9}}{(1-x)^{10}}}\,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$
10 Centered decacross

10-orthoplex

210 9-cell

(20, ..., 1024)

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

${\displaystyle {\frac {(1+x)^{10}}{(1-x)^{11}}}\,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$
11 Centered hendecacross

11-orthoplex

211 10-cell

(22, ..., 2048)

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

${\displaystyle {\frac {(1+x)^{11}}{(1-x)^{12}}}\,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$
12 Centered dodecacross

12-orthoplex

212 11-cell

(24, ..., 4096)

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

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

Table of sequences

Centered orthoplicial polytopic numbers sequences
d ${\displaystyle \,_{c}P_{2d}^{(d)}(n)\,}$ 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, 93, 95, 97, 99, 101, 103, 105, ...}
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, 2113, 2245, 2381, 2521, 2665, ...}
3 {1, 7, 25, 63, 129, 231, 377, 575, 833, 1159, 1561, 2047, 2625, 3303, 4089, 4991, 6017, 7175, 8473, 9919, 11521, 13287, 15225, 17343, 19649, 22151, 24857, 27775, 30913, 34279, 37881, 41727, ...}
4 {1, 9, 41, 129, 321, 681, 1289, 2241, 3649, 5641, 8361, 11969, 16641, 22569, 29961, 39041, 50049, 63241, 78889, 97281, 118721, 143529, 172041, 204609, 241601, 283401, 330409, 383041, 441729, ...}
5 {1, 11, 61, 231, 681, 1683, 3653, 7183, 13073, 22363, 36365, 56695, 85305, 124515, 177045, 246047, 335137, 448427, 590557, 766727, 982729, 1244979, 1560549, 1937199, 2383409, 2908411, ...}
6 {1, 13, 85, 377, 1289, 3653, 8989, 19825, 40081, 75517, 134245, 227305, 369305, 579125, 880685, 1303777, 1884961, 2668525, 3707509, 5064793, 6814249, 9041957, 11847485, 15345233, 19665841, ...}
7 {1, 15, 113, 575, 2241, 7183, 19825, 48639, 108545, 224143, 433905, 795455, 1392065, 2340495, 3800305, 5984767, 9173505, 13726991, 20103025, 28875327, 40754369, 56610575, 77500017, ...}
8 {1, 17, 145, 833, 3649, 13073, 40081, 108545, 265729, 598417, 1256465, 2485825, 4673345, 8405905, 14546705, 24331777, 39490049, 62390545, 96220561, 145198913, 214828609, 312193553, ...}
9 {1, 19, 181, 1159, 5641, 22363, 75517, 224143, 598417, 1462563, 3317445, 7059735, 14218905, 27298155, 50250765, 89129247, 152951073, 254831667, 413442773, 654862247, 1014889769, ...}
10 {1, 21, 221, 1561, 8361, 36365, 134245, 433905, 1256465, 3317445, 8097453, 18474633, 39753273, 81270333, 158819253, 298199265, 540279585, 948062325, 1616336765, 2684641785, 4354393801, ...}
11 {1, 23, 265, 2047, 11969, 56695, 227305, 795455, 2485825, 7059735, 18474633, 45046719, 103274625, 224298231, 464387817, 921406335, 1759885185, 3248227095, 5812626185, 10113604735, ...}
12 {1, 25, 313, 2625, 16641, 85305, 369305, 1392065, 4673345, 14218905, 39753273, 103274625, 251595969, 579168825, 1267854873, 2653649025, 5334940545, 10343052825, 19403906105, ...}