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Centered regular polychoron numbers
A001846 Centered 4-dimensional orthoplex numbers.
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Contents
Formulae
The nth 4-dimensional N3-cell centered regular polytope (having N0 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 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 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 k-gon numbers forms a basis of order , i.e. every nonnegative integer can be written as a sum of 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 such that every nonnegative integer is a sum of th powers, i.e. the set of th powers forms a basis of order . The Hilbert-Waring 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
N0, N1, N2 and N3 are the number of vertices (0-dimensional), edges (1-dimensional), faces (2-dimensional) and cells (3-dimensional) respectively, where the cells are the actual facets. The centered regular polychorons are listed by increasing number N0 of vertices.
Rank
|
N0 | Name
(N0, N1, N2, N3) 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 |
N0, N1, N2 and N3 are the number of vertices (0-dimensional), edges (1-dimensional), faces (2-dimensional) and cells (3-dimensional) respectively, where the cells are the actual facets. The centered regular polychorons are listed by increasing number N0 of vertices.
Rank
|
N0 | Name
(N0, N1, N2, N3) 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
N0 | 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 nth d-dimensional centered regular convex polytope number with N0 0-dimensional elements (vertices V.)
- ↑ Weisstein, Eric W., Fermat's Polygonal Number Theorem, From MathWorld--A Wolfram Web Resource.
- ↑ 3.0 3.1 Weisstein, Eric W., Schläfli Symbol, From MathWorld--A 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, 2nd ed., 1994.