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Centered regular polychoron numbers
A001846 Centered 4dimensional orthoplex numbers.
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Contents
Formulae
The n^{th} 4dimensional N_{3}cell centered regular polytope (having N_{0} vertices) number is given by the formula:^{[1]}
where ...
Recurrence equations
with initial conditions
Generating function
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 kpolygonal 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 kgon numbers (known as the polygonal number theorem,) while a vertical (higher dimensional) generalization has also been made (known as the HilbertWaring problem.)
A nonempty subset of nonnegative integers is called a basis of order if is the minimum number with the property that every nonnegative integer can be written as a sum of elements in . Lagrange’s sum of four squares can be restated as the set of nonnegative squares forms a basis of order 4.
Theorem (Cauchy) For every , the set of kgon numbers forms a basis of order , i.e. every nonnegative integer can be written as a sum of kgon 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 such that every nonnegative integer is a sum of ^{th} powers, i.e. the set of ^{th} powers forms a basis of order . The HilbertWaring problem is concerned with the study of for . 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
Partial sums
Partial sums of reciprocals
Sum of reciprocals
Table of formulae and values
N_{0}, N_{1}, N_{2} and N_{3} are the number of vertices (0dimensional), edges (1dimensional), faces (2dimensional) and cells (3dimensional) respectively, where the cells are the actual facets. The centered regular polychorons are listed by increasing number N_{0} of vertices.
Rank

N_{0}  Name
(N_{0}, N_{1}, N_{2}, N_{3}) Schläfli symbol^{[3]} 
Formulae

Generating
function

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

0  5  Pentachoron
5 cell (5, 10, 10, 5) {3, 3, 3} 
1  6  
1  8  16 cell
(8, 24, 32, 16) {3, 3, 4} 
1  9  
2  16  Tesseract
8 cell (16, 32, 24, 8) {4, 3, 3} 
1  17  
3  24  24 cell
(24, 96, 96, 24) {3, 4, 3} 
1  25  
4  120  600 cell
(120, 720, 1200, 600) {3, 3, 5} 
1  121  
5  600  120 cell
(600, 1200, 720, 120) {5, 3, 3} 
1  601 
N_{0}, N_{1}, N_{2} and N_{3} are the number of vertices (0dimensional), edges (1dimensional), faces (2dimensional) and cells (3dimensional) respectively, where the cells are the actual facets. The centered regular polychorons are listed by increasing number N_{0} of vertices.
Rank

N_{0}  Name
(N_{0}, N_{1}, N_{2}, N_{3}) Schläfli symbol^{[3]} 
Order
of basis

Differences

Partial sums

Partial sums of reciprocals

Sum of Reciprocals^{[4]}^{[5]}


0  5  Pentachoron
5 cell (5, 10, 10, 5) {3, 3, 3} 

1  8  16 cell
(8, 24, 32, 16) {3, 3, 4} 

2  16  Tesseract
8 cell (16, 32, 24, 8) {4, 3, 3} 

3  24  24 cell
(24, 96, 96, 24) {3, 4, 3} 

4  120  600 cell
(120, 720, 1200, 600) {3, 3, 5} 

5  600  120 cell
(600, 1200, 720, 120) {5, 3, 3} 
Table of sequences
N_{0}  sequences 

5  {1, 6, 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, ...} 
8  {1, 9, ...} 
16  {1, 17, ...} 
24  {1, 25, ...} 
120  {1, 121, ...} 
600  {1, 601, ...} 
See also
Notes
 ↑ Where is the n^{th} ddimensional centered regular convex polytope number with N_{0} 0dimensional elements (vertices V.)
 ↑ Weisstein, Eric W., Fermat's Polygonal Number Theorem, From MathWorldA Wolfram Web Resource.
 ↑ ^{3.0} ^{3.1} Weisstein, Eric W., Schläfli Symbol, From MathWorldA Wolfram Web Resource.
 ↑ Downey, Lawrence M., Ong, Boon W., and Sellers, James A., Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, 2008.
 ↑ PSYCHEDELIC GEOMETRY, INVERSE POLYGONAL NUMBERS SERIES.
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.