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Square hyperpyramidal numbers
The simplicial polytopic numbers are triangular (hyper)pyramidal numbers, starting with a triangle for dimension d = 2, then do a pyramidal stacking for each extra dimension.
The square (hyper)pyramidal numbers are starting with a square for dimension d = 2, then do a pyramidal stacking for each extra dimension.
While the simplicial polytopic numbers correspond to regular polytopes (regular simplex polytopes), the square (hyper)pyramidal numbers, for d ≥ 3, correspond to nonregular polytopes (square (hyper)pyramids.) They are nonetheless especially interesting since they are the building blocks for the orthoplicial polytopic numbers, which are regular polytopes (regular othoplex polytopes.)
All figurate numbers are accessible via this structured menu: Classifications of figurate numbers
Contents
- 1 (2,1)-Pascal triangle or (1,2)-Pascal triangle and square (hyper)pyramidal numbers
- 2 Formulae
- 3 Descartes-Euler (convex) polytope formula
- 4 Recurrence equation
- 5 Generating function
- 6 Order of basis
- 7 Differences
- 8 Partial sums
- 9 Partial sums of reciprocals
- 10 Sum of reciprocals
- 11 Table of formulae and values
- 12 Table of related formulae and values
- 13 Table of sequences
- 14 See also
- 15 Notes
- 16 External links
(2,1)-Pascal triangle or (1,2)-Pascal triangle and square (hyper)pyramidal numbers
Formulae
Descartes-Euler (convex) polytope formula
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 square hyperpyramidal numbers are listed by increasing number N0 of vertices.
d | (N0, N1, N2, ...)
Schläfli symbol[1] |
Formulae
|
n = 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | OEIS
number |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | Square gnomon
(2) {} |
|
0 | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | A004273(n) |
2 | Square
(3, 3) {3} |
|
0 | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 | A000290(n) |
3 | Square pyramidal
(, , ) {, } |
|
0 | 1 | 5 | 14 | 30 | 55 | 91 | 140 | 204 | 285 | 385 | 506 | 650 | A000330(n) |
4 |
(, , , ) {, , } |
0 | 1 | 6 | 20 | 50 | 105 | 196 | 336 | 540 | 825 | 1210 | 1716 | 2366 | A002415(n+1) | |
5 |
(, , , , ) {, , , } |
0 | 1 | 7 | 27 | 77 | 182 | 378 | 714 | 1254 | 2079 | 3289 | 5005 | 7371 | A005585(n) | |
6 |
(, , , , , ) {, , , , } |
0 | 1 | 8 | 35 | 112 | 294 | 672 | 1386 | 2640 | 4719 | 8008 | 13013 | 20384 | A040977(n-1) | |
7 |
(, , , , , , ) {, , , , , } |
0 | 1 | 9 | 44 | 156 | 450 | 1122 | 2508 | 5148 | 9867 | 17875 | 30888 | 51272 | A050486(n-1) | |
8 |
(, , , , , , , ) {, , , , , , } |
0 | 1 | 10 | 54 | 210 | 660 | 1782 | 4290 | 9438 | 19305 | 37180 | 68068 | 119340 | A053347(n-1) | |
9 |
(, , , , , , , , ) {, , , , , , , } |
0 | 1 | 11 | 65 | 275 | 935 | 2717 | 7007 | 16445 | 35750 | 72930 | 140998 | 260338 | A054333(n-1) | |
10 |
(, , , , , , , , , ) {, , , , , , , , } |
0 | 1 | 12 | 77 | 352 | 1287 | 4004 | 11011 | 27456 | 63206 | 136136 | 277134 | 537472 | A054334(n-1) | |
11 |
(, , , , , , , , , , ) {, , , , , , , , , } |
0 | 1 | 13 | 90 | 442 | 1729 | 5733 | 16744 | 44200 | 107406 | 243542 | 520676 | 1058148 | A057788(n-1) | |
12 |
(, , , , , , , , , , , ) {, , , , , , , , , , } |
0 | 1 | 14 | 104 | 546 | 2275 | 8008 | 24752 | 68952 | 176358 | 419900 | 940576 | 1998724 | Axxxxxx |
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 square hyperpyramidal numbers are listed by increasing number N0 of vertices.
d | Generating
function
|
Order
of basis |
Differences
|
Partial sums
|
Partial sums of reciprocals
|
Sum of reciprocals[6]
|
---|---|---|---|---|---|---|
1 | ||||||
2 | ||||||
3 |
|
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4 | ||||||
5 | ||||||
6 | ||||||
7 | ||||||
8 | ||||||
9 | ||||||
10 | ||||||
11 | ||||||
12 |
Table of sequences
d | Sequence |
---|---|
1 | {0, 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, 107, ...} |
2 | {0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, ...} |
3 | {0, 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, 1015, 1240, 1496, 1785, 2109, 2470, 2870, 3311, 3795, 4324, 4900, 5525, 6201, 6930, 7714, 8555, 9455, 10416, 11440, 12529, 13685, 14910, ...} |
4 | {0, 1, 6, 20, 50, 105, 196, 336, 540, 825, 1210, 1716, 2366, 3185, 4200, 5440, 6936, 8721, 10830, 13300, 16170, 19481, 23276, 27600, 32500, 38025, 44226, 51156, 58870, 67425, 76880, 87296, 98736, ...} |
5 | {0, 1, 7, 27, 77, 182, 378, 714, 1254, 2079, 3289, 5005, 7371, 10556, 14756, 20196, 27132, 35853, 46683, 59983, 76153, 95634, 118910, 146510, 179010, 217035, 261261, 312417, 371287, 438712, ...} |
6 | {0, 1, 8, 35, 112, 294, 672, 1386, 2640, 4719, 8008, 13013, 20384, 30940, 45696, 65892, 93024, 128877, 175560, 235543, 311696, 407330, 526240, 672750, 851760, 1068795, 1330056, 1642473, ... } |
7 | {0, 1, 9, 44, 156, 450, 1122, 2508, 5148, 9867, 17875, 30888, 51272, 82212, 127908, 193800, 286824, 415701, 591261, 826804, 1138500, 1545830, 2072070, 2744820, 3596580, 4665375, 5995431, ...} |
8 | {0, 1, 10, 54, 210, 660, 1782, 4290, 9438, 19305, 37180, 68068, 119340, 201552, 329460, 523260, 810084, 1225785, 1817046, 2643850, 3782350, 5328180, 7400250, 10145070, 13741650, 18407025, ...} |
9 | {0, 1, 11, 65, 275, 935, 2717, 7007, 16445, 35750, 72930, 140998, 260338, 461890, 791350, 1314610, 2124694, 3350479, 5167525, 7811375, 11593725, 16921905, 24322155, 34467225, 48208875, ...} |
10 | {0, 1, 12, 77, 352, 1287, 4004, 11011, 27456, 63206, 136136, 277134, 537472, 999362, 1790712, 3105322, 5230016, 8580495, 13748020, 21559395, 33153120, 50075025, 74397180, 108864405, ...} |
11 | {0, 1, 13, 90, 442, 1729, 5733, 16744, 44200, 107406, 243542, 520676, 1058148, 2057510, 3848222, 6953544, 12183560, 20764055, 34512075, 56071470, 89224590, 139299615, 213696795, ...} |
12 | {0, 1, 14, 104, 546, 2275, 8008, 24752, 68952, 176358, 419900, 940576, 1998724, 4056234, 7904456, 14858000, 27041560, 47805615, 82317690, 138389160, 227613750, 366913365, 580610160, ...} |
See also
Centered square hyperpyramidal numbers, i.e. (Centered squares) hyperpyramidal numbers.
Notes
- ↑ Weisstein, Eric W., Schläfli Symbol, From MathWorld--A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Rising Factorial, 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.