Centered orthoplex numbers

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

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Formulae

The n^th d-dimensional centered orthoplicial polytopic number is given by the formula:

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

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:

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

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

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

with initial conditions:

$\,_{c}P^{(d)}(2d,0)=1,\,$

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

Generating function

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

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. 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 $\scriptstyle k\geq 3$, 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. 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 $\scriptstyle d\geq 2\,$. 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

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

Partial sums

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

where $\scriptstyle t_{m}\,$ is the m^th triangular number.

Partial sums of reciprocals

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

Sum of reciprocals

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

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	Formulae $\,_{c}P_{2d}^{(d)}(n)=?\,$	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) {}	$n+(n+1)\,$ $2n+1\,$ Odd numbers	1	3	5	7	9	11	13	15	17	19	21	23	25	A005408
2	Centered square 2-orthoplex Tetragon Bicross (4, 4) {4}	$4T_{n}+1\,$ $2n(n+1)+1\,$ $n^{2}+(n+1)^{2}\,$ $1+2n+2n^{2}\,$	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}	${\frac {(2n+1)(2n^{2}+2n+3)}{3}}\,$ ${\frac {3+8n+6n^{2}+4n^{3}}{3}}\,$	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}	${\frac {3+8n+10n^{2}+4n^{3}+2n^{4}}{3}}\,$	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}	$\scriptstyle {\frac {15+46n+50n^{2}+40n^{3}+10n^{4}+4n^{5}}{15}}\,$ $\scriptstyle {\frac {(2n+1)(2n(n+1)(n^{2}+n+8)+15)}{15}}\,$	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}	$\scriptstyle {\frac {45+138n+196n^{2}+120n^{3}+70n^{4}+12n^{5}+4n^{6}}{45}}\,$	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}	$\scriptstyle {\frac {315+1056n+1372n^{2}+1232n^{3}+490n^{4}+224n^{5}+28n^{6}+8n^{7}}{315}}\,$	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}	$\scriptstyle {\frac {315+1056n+1636n^{2}+1232n^{3}+798n^{4}+224n^{5}+84n^{6}+8n^{7}+2n^{8}}{315}}\,$	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}	$\scriptstyle {\frac {2835+10134n+14724n^{2}+14360n^{3}+7182n^{4}}{2835}}\,+\,$ $\scriptstyle {\frac {3612n^{5}+756n^{6}+240n^{7}+18n^{8}+4n^{9}}{2835}}\,$	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}	$\scriptstyle {\frac {14175+50670n+83754n^{2}+71800n^{3}+50270n^{4}}{14175}}\,+\,$ $\scriptstyle {\frac {18060n^{5}+7392n^{6}+1200n^{7}+330n^{8}+20n^{9}+4n^{10}}{14175}}\,$	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}	$\scriptstyle {\frac {155925+585720n+921294n^{2}+957308n^{3}+552970n^{4}+299200n^{5}}{155925}}\,+\,$ $\scriptstyle {\frac {81312n^{6}+27984n^{7}+3630n^{8}+880n^{9}+44n^{10}+8n^{11}}{155925}}\,$	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}	$\scriptstyle {\frac {467775+1757160n+3056742n^{2}+2871924n^{3}+2137564n^{4}+897600n^{5}}{467775}}\,+\,$ $\scriptstyle {\frac {393536n^{6}+83952n^{7}+24882n^{8}+2640n^{9}+572n^{10}+24n^{11}+4n^{12}}{467775}}\,$	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	Generating function $G_{\,_{c}P_{2d}^{(d)}}(x)=\,$ ${\frac {(1+x)^{d}}{(1-x)^{d+1}}}\,$	Order of basis $g_{\,_{c}P_{2d}^{(d)}}\,$	Differences $\,_{c}P_{2d}^{(d)}(n)-\,$ $\,_{c}P_{2d}^{(d)}(n-1)=\,$ $?\,$	Partial sums $\sum _{n=0}^{m}{\,_{c}P_{2d}^{(d)}(n)}=?\,$	Partial sums of reciprocals $\sum _{n=0}^{m}{1 \over {\,_{c}P_{2d}^{(d)}(n)}}=?\,$	Sum of reciprocals $\sum _{n=0}^{\infty }{1 \over {\,_{c}P_{2d}^{(d)}(n)}}=?\,$
1	Centered square gnomon 1-orthoplex di-0-cell (2) {}	${\frac {(1+x)}{(1-x)^{2}}}\,$	$2\,$	$2\,$	$(m+1)^{2}\,$	${\frac {(\psi (m+{\tfrac {3}{2}})+\gamma )}{2}}+\log(2)\,$	$\infty \,$
2	Centered square 2-orthoplex Tetra-1-cell (4, 4) {4}	${(1+x)^{2}} \over {(1-x)^{3}}\,$ ${x^{2}+2x+1} \over {(1-x)^{3}}\,$	$\,$	$4n\,$	${\frac {(m+1)(2m^{2}+4m+3)}{3}}\,$	$\,$	${\frac {\pi }{2}}\tanh {\bigg (}{\frac {\pi }{2}}{\bigg )}-1\,$
3	Centered octahedral 3-orthoplex Octa-2-cell (6, 12, 8) {3, 4}	${\frac {(1+x)^{3}}{(1-x)^{4}}}\,$ ${\frac {(1+x)(1+2x+x^{2})}{(1-x)^{4}}}\,$	$\,$	$\,$	${\frac {(m+1)^{2}(m^{2}+2m+3)}{3}}\,$	$\,$	$\,$
4	Centered tetracross 4-orthoplex 24 3-cell (8, 24, 32, 16) {3, 3, 4}	${\frac {(1+x)^{4}}{(1-x)^{5}}}\,$	$\,$	$\,$	$\,$	$\,$	$\,$
5	Centered pentacross 5-orthoplex 25 4-cell (10, 40, 80, 80, 32) {3, 3, 3, 4}	${\frac {(1+x)^{5}}{(1-x)^{6}}}\,$	$\,$	$\,$	$\,$	$\,$	$\,$
6	Centered hexacross 6-orthoplex 26 5-cell (12, 60, 160, 240, 192, 64) {3, 3, 3, 3, 4}	${\frac {(1+x)^{6}}{(1-x)^{7}}}\,$	$\,$	$\,$	$\,$	$\,$	$\,$
7	Centered heptacross 7-orthoplex 27 6-cell (14, 84, 280, 560, 672, 448, 128) {3, 3, 3, 3, 3, 4}	${\frac {(1+x)^{7}}{(1-x)^{8}}}\,$	$\,$	$\,$	$\,$	$\,$	$\,$
8	Centered octacross 8-orthoplex 28 7-cell (16, 112, 448, 1120, 1792, 1792, 1024, 256) {3, 3, 3, 3, 3, 3, 4}	${\frac {(1+x)^{8}}{(1-x)^{9}}}\,$	$\,$	$\,$	$\,$	$\,$	$\,$
9	Centered enneacross 9-orthoplex 29 8-cell (18, ..., 512) {3, 3, 3, 3, 3, 3, 3, 4}	${\frac {(1+x)^{9}}{(1-x)^{10}}}\,$	$\,$	$\,$	$\,$	$\,$	$\,$
10	Centered decacross 10-orthoplex 210 9-cell (20, ..., 1024) {3, 3, 3, 3, 3, 3, 3, 3, 4}	${\frac {(1+x)^{10}}{(1-x)^{11}}}\,$	$\,$	$\,$	$\,$	$\,$	$\,$
11	Centered hendecacross 11-orthoplex 211 10-cell (22, ..., 2048) {3, 3, 3, 3, 3, 3, 3, 3, 3, 4}	${\frac {(1+x)^{11}}{(1-x)^{12}}}\,$	$\,$	$\,$	$\,$	$\,$	$\,$
12	Centered dodecacross 12-orthoplex 212 11-cell (24, ..., 4096) {3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4}	${\frac {(1+x)^{12}}{(1-x)^{13}}}\,$	$\,$	$\,$	$\,$	$\,$	$\,$

Table of sequences

Centered orthoplicial polytopic numbers sequences

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

See also

Orthoplicial polytopic 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.