Centered regular orthotopic numbers

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

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Formulae

The n^th d-dimensional centered regular orthotopic number is given by the formula:

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

where d is the dimension and 2^d 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

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

with initial conditions

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

Generating function

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

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

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

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

where $\scriptstyle A_{d}(x)\,$ is the d^th, 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. 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

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

Partial sums

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

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

Partial sums of reciprocals

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

Sum of reciprocals

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

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	Formulae $\,_{c}P_{2^{d}}^{(d)}(n)=\,$ $\scriptstyle n^{d}+(n+1)^{d}\,$	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) {}	$\scriptstyle n+(n+1)\,$ $\scriptstyle 2n+1\,$ Odd numbers	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}	$\scriptstyle n^{2}+(n+1)^{2}\,$ $\scriptstyle 2n(n+1)+1\,$ $\scriptstyle 4T_{n}+1\,$	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 tetradeca-6-cell (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 hexadeca-7-cell (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 octadeca-8-cell (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	Generating function $G_{\{\,_{c}P_{2^{d}}^{(d)}(n)\}}(x)=\,$ $\scriptstyle {{\big (}{\frac {1+x}{x}}{\big )}\,G_{\{P_{2^{d}}^{(d)}\}}(x)\,=\,{\big (}{\frac {1+x}{1-x}}{\big )}\sum _{i=0}^{d-1}{\binom {d}{i}}G(x^{i})}\,$ $\scriptstyle {\frac {(1+x)A_{d}(x)}{(1-x)^{d+1}}}\,$ $\scriptstyle {{(1+x)(1+(2^{d}-d-1)x+...+(2^{d}-d-1)x^{d-2}+x^{d-1})} \over {(1-x)^{d+1}}}\,$ where $\scriptstyle A_{d}(x)\,$ is the dth, d ≥ 0, Eulerian polynomial.	Order of basis $g_{\{\,_{c}P_{2^{d}}^{(d)}\}}\,$	Differences $\,_{c}P_{2^{d}}^{(d)}(n)-\,$ $\,_{c}P_{2^{d}}^{(d)}(n-1)=\,$ $\scriptstyle (n+1)^{d}-(n-1)^{d}\,$	Partial sums $\sum _{n=1}^{m}{\,_{c}P_{2^{d}}^{(d)}(n)}=\,$	Partial sums of reciprocals $\sum _{n=1}^{m}{1 \over {\,_{c}P_{2^{d}}^{(d)}(n)}}=\,$	Sum of reciprocals $\sum _{n=1}^{\infty }{1 \over {P_{2^{d}}^{(d)}(n)}}=\,$
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 tetradeca-6-cell (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 hexadeca-7-cell (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 octadeca-8-cell (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	$\,_{c}P_{2d}^{(d)}(n),\ n\geq 0\,$ 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, ...}

See also

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.