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(Centered squares) hyperpyramidal numbers
The (centered squares) hyperpyramidal numbers are starting with a centered square for dimension d = 2, then do a pyramidal stacking for each extra dimension.
Although the Simplicial polytopic numbers are triangular hyperpyramidal numbers, the centered simplicial polytopic numbers are NOT (centered triangles) hyperpyramidal numbers.
All figurate numbers are accessible via this structured menu: Classifications of figurate numbers
Contents
- 1 Formulae
- 2 Schläfli-Poincaré (convex) polytope formula
- 3 Recurrence equation
- 4 Generating function
- 5 Order of basis
- 6 Differences
- 7 Partial sums
- 8 Partial sums of reciprocals
- 9 Sum of reciprocals
- 10 Table of formulae and values
- 11 Table of related formulae and values
- 12 Table of sequences
- 13 See also
- 14 Notes
- 15 External links
Formulae
Schläfli-Poincaré (convex) polytope formula
Schläfli-Poincaré generalization of the Descartes-Euler (convex) polyhedral formula.[2]
For d-dimensional convex polytopes:
where N0 is the number of 0-dimensional elements (vertices V), N1 is the number of 1-dimensional elements (edges V), N2 is the number of 2-dimensional elements (faces F) and N3 is the number of 3-dimensional elements (cells C), and so on... of the convex polytope.
Recurrence equation
Generating function
Order of basis
Differences
Partial sums
Partial sums of reciprocals
Sum of reciprocals
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 squares) hyperpyramidal numbers are listed by increasing number N0 of vertices.
d | Name
(N0, N1, N2, ...) Schläfli symbol[3] |
Formulae
|
n = 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | OEIS
number |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3 | Centered square pyramidal
(, , ) {, } |
|
0 | 1 | 6 | 19 | 44 | 85 | 146 | 231 | 344 | 489 | 670 | 891 | 1156 | A005900(n) |
4 |
(, , , ) {, , } |
1 | ||||||||||||||
5 |
(, , , , ) {, , , } |
1 | ||||||||||||||
6 |
(, , , , , ) {, , , , } |
1 | ||||||||||||||
7 |
(, , , , , , ) {, , , , , } |
1 | ||||||||||||||
8 |
(, , , , , , , ) {, , , , , , } |
1 | ||||||||||||||
9 |
(, , , , , , , , ) {, , , , , , , } |
1 | ||||||||||||||
10 |
(, , , , , , , , , ) {, , , , , , , , } |
1 | ||||||||||||||
11 |
(, , , , , , , , , , ) {, , , , , , , , , } |
1 | ||||||||||||||
12 |
(, , , , , , , , , , , ) {, , , , , , , , , , } |
1 |
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 elements are the actual facets. The (centered squares) hyperpyramidal numbers are listed by increasing number N0 of vertices.
d | Name
(N0, N1, N2, ...) Schläfli symbol[3] |
Generating
function
|
Order
of basis |
Differences
|
Partial sums
|
Partial sums of reciprocals
|
Sum of reciprocals[7]
|
---|---|---|---|---|---|---|---|
3 | |||||||
4 | |||||||
5 | |||||||
6 | |||||||
7 | |||||||
8 | |||||||
9 | |||||||
10 | |||||||
11 | |||||||
12 |
Table of sequences
d | sequences |
---|---|
3 | {0, 1, 6, 19, 44, 85, 146, 231, 344, 489, 670, 891, 1156, 1469, 1834, 2255, 2736, 3281, 3894, 4579, 5340, 6181, 7106, 8119, 9224, 10425, 11726, 13131, 14644, 16269, ...} |
4 | {0, 1, ...} |
5 | {0, 1, ...} |
6 | {0, 1, ...} |
7 | {0, 1, ...} |
8 | {0, 1, ...} |
9 | {0, 1, ...} |
10 | {0, 1, ...} |
11 | {0, 1, ...} |
12 | {0, 1, ...} |
See also
Notes
- ↑ Where , k ≥ 1, n ≥ 0, is the d-dimensional, d ≥ 0, (centered (k+2)-gonal base) (hyper)pyramidal number where, for d ≥ 2, is the number of vertices (including the apex vertices) of the (centered polygonal base) (hyper)pyramid (the quoted emphasizes that only the polygons are centered, not the whole figure.)
- ↑ Weisstein, Eric W., Polyhedral Formula, From MathWorld--A Wolfram Web Resource.
- ↑ 3.0 3.1 Weisstein, Eric W., Schläfli Symbol, From MathWorld--A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Fermat's Polygonal Number Theorem, From MathWorld--A Wolfram Web Resource.
- ↑ HYUN KWANG KIM, ON REGULAR POLYTOPE NUMBERS.
- ↑ Pollock, Frederick, On the extension of the principle of Fermat's theorem on the polygonal numbers to the higher order of series whose ultimate differences are constant. With a new theorem proposed, applicable to all the orders, Abstracts of the Papers Communicated to the Royal Society of London, 5 (1850) pp. 922-924.
- ↑ Downey, Lawrence M., Ong, Boon W., and Sellers, James A., Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, 2008.
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.