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Regular orthotopic numbers
d |
d |
d |
d > 3 |
d |
All figurate numbers are accessible via this structured menu: Classifications of figurate numbers
Contents
- 1 Formulae
- 2 Recurrence relation
- 3 Generating function
- 4 Order of basis
- 5 Differences
- 6 Partial sums
- 7 Partial sums of reciprocals
- 8 Sum of reciprocals
- 9 Number of j-dimensional “vertices”
- 10 Table of formulae and values
- 11 Table of related formulae and values
- 12 Table of sequences
- 13 Regular orthotopic numbers read cross-dimensionally (giving exponentials sequences)
- 14 See also
- 15 Notes
- 16 External links
Formulae
Then |
d |
d |
d |
Recurrence relation
Generating function
where
d |
can be recursively generated with the triangle of Eulerian numbers.
- with
Method for obtaining the generating functions for successive powers
Since , the generating function of 1 is then[3] [4]
Since , the generating function of is then
- and which gives
Since , the generating function of is then
- and which gives
Since , the generating function of is then
- and which gives
Since , the generating function of is then
- and which gives
Since , the generating function of is then
- and which gives
And the general recurrence equation for the generating function of powers is
- with
Simpler method for obtaining the generating functions for successive powers
Since , we have
then
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, andk |
k |
A |
g |
g |
g |
A |
k ≥ 3 |
k |
k |
k |
g (d) |
d ≥ 2 |
The (presumed) solution to Waring’s problem is (see A002804)
with series representation
- for and .
n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 4 | 9 | 19 | 37 | 73 | 143 | 279 | 548 | 1079 | 2132 | 4223 | 8384 | 16673 | 33203 |
Differences
Partial sums
Partial sums of reciprocals
Sum of reciprocals
Number of j-dimensional “vertices”
Table of formulae and values
N0, N1, N2, N3, ... |
d − 1 |
N0 |
|
Name Regular
(
Schläfli symbol[10] |
Formulae
|
|
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | A-number | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0 | Point numbers
0-orthotope zero-(-1)-cell () {} |
|
0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | (NOT | |||||||
1 | Segment numbers
1-orthotope di-0-cell (2) {} |
|
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | A001477 | |||||||
2 | Square numbers
2-orthotope tetra-1-cell (4, 4) {4} |
|
0 | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 | A000290 | |||||||
3 | Cube numbers
3-orthotope hexa-2-cell (8, 12, 6) {4, 3} |
|
0 | 1 | 8 | 27 | 64 | 125 | 216 | 343 | 512 | 729 | 1000 | 1331 | 1728 | A000578 | |||||||
4 | Tesseract numbers
4-orthotope octa-3-cell (16, 32, 24, 8) {4, 3, 3} |
|
0 | 1 | 16 | 81 | 256 | 625 | 296 | 2401 | 4096 | 6561 | 10000 | 14641 | 20736 | A000583 | |||||||
5 | Penteract numbers
5-orthotope deca-4-cell (32, 80, 80, 40, 10) {4, 3, 3, 3} |
|
0 | 1 | 32 | 243 | 1024 | 3125 | 7776 | 16807 | 32768 | 59049 | 100000 | 161051 | 248832 | A000584 | |||||||
6 | Hexeract numbers
6-orthotope dodeca-5-cell (64, 192, 240, 160, 60, 12) {4, 3, 3, 3, 3} |
|
0 | 1 | 64 | 729 | 4096 | 15625 | 46656 | 117649 | 262144 | 531441 | 1000000 | 1771561 | 2985984 | A001014 | |||||||
7 | Hepteract numbers
7-orthotope tetradeca-6-cell (128, 448, 672, 560, 280, 84, 14) {4, 3, 3, 3, 3, 3} |
|
0 | 1 | 128 | 2187 | 16384 | 78125 | 279936 | 823543 | 2097152 | 4782969 | 10000000 | 19487171 | 35831808 | A001015 | |||||||
8 | Octeract numbers
8-orthotope hexadeca-7-cell (256, 1024, 1792, 1792, 1120, 448, 112, 16) {4, 3, 3, 3, 3, 3, 3} |
|
0 | 1 | 256 | 6561 | 65536 | 390625 | 1679616 | 5764801 | 16777216 | 43046721 | 100000000 | 214358881 | 429981696 | A001016 | |||||||
9 | Enneract numbers
9-orthotope octadeca-8-cell (512, 2304, 4608, 5376, 4032, 2016, 672, 144, 18) {4, 3, 3, 3, 3, 3, 3, 3} |
|
0 | 1 | 512 | 19683 | 262144 | 1953125 | 10077696 | 40353607 | 134217728 | 387420489 | 1000000000 | 2357947691 | 5159780352 | A001017 | |||||||
10 | 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} |
|
0 | 1 | 1024 | 59049 | 1048576 | 9765625 | 60466176 | 282475249 | 1073741824 | 3486784401 | 10000000000 | 25937424601 | 61917364224 | A008454 | |||||||
11 | Hendekeract numbers
11-orthotope icosidi-10-cell (..., 22) {4, 3, 3, 3, 3, 3, 3, 3, 3, 3} |
|
0 | 1 | 2048 | 177147 | 4194304 | 48828125 | 362797056 | 1977326743 | 8589934592 | 31381059609 | 100000000000 | 285311670611 | 743008370688 | A008455 | |||||||
12 | Dodekeract numbers
12-orthotope icositetra-11-cell (..., 24) {4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3} |
|
0 | 1 | 4096 | 531441 | 16777216 | 244140625 | 2176782336 | 13841287201 | 68719476736 | 282429536481 | 1000000000000 | 3138428376721 | 8916100448256 | A008456 |
N0, N1, N2, N3, ... |
d − 1 |
N0 |
Name
Regular -orthotope -cell
|
Generating
function
|
Order
of basis
|
Differences[12]
|
Partial sums
|
Partial sums of reciprocals
|
Sum of reciprocals[14]
| |
---|---|---|---|---|---|---|---|
0 | Point numbers
0-orthotope zero-(-1)-cell () {} |
|
|
|
|
| |
1 | Segment numbers
1-orthotope di-0-cell (2) {} |
|
[2]
|
| |||
2 | Square numbers
2-orthotope tetra-1-cell (4, 4) {4} |
|
|
| |||
3 | Cube numbers
3-orthotope hexa-2-cell (8, 12, 6) {4, 3} |
|
Hex (or centered hexagonal) numbers[19] |
|
[20] | ||
4 | Tesseract numbers
4-orthotope octa-3-cell (16, 32, 24, 8) {4, 3, 3} |
|
Rhombic dodecahedral numbers [21] |
|
| ||
5 | Penteract numbers
5-orthotope deca-4-cell (32, 80, 80, 40, 10) {4, 3, 3, 3} |
|
|
| |||
6 | Hexeract numbers
6-orthotope dodeca-5-cell (64, 192, 240, 160, 60, 12) {4, 3, 3, 3, 3} |
|
| ||||
7 | Hepteract numbers
7-orthotope tetradeca-6-cell (128, 448, 672, 560, 280 , 84, 14) {4, 3, 3, 3, 3, 3} |
|
| ||||
8 | Octeract numbers
8-orthotope hexadeca-7-cell (256, 1024 , 1792, 1792, 1120, 448, 112, 16) {4, 3, 3, 3, 3, 3, 3} |
|
| ||||
9 | Enneract numbers
9-orthotope octadeca-8-cell (..., 18) {4, 3, 3, 3, 3, 3, 3, 3} |
|
| ||||
10 | Dekeract numbers
10-orthotope icosa-9-cell (..., 20) {4, 3, 3, 3, 3, 3, 3, 3, 3} |
|
|||||
11 | Hendekeract numbers
11-orthotope icosidi-10-cell (..., 22) {4, 3, 3, 3, 3, 3, 3, 3, 3, 3} |
|
| ||||
12 | Dodekeract numbers
12-orthotope icositetra-11-cell (..., 24) {4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3} |
|
Table of sequences
|
A-number |
| ||
---|---|---|---|---|
0 | (NOT |
{0, 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, 1, 1, 1, 1, 1, 1, 1, ...} | ||
1 | A001477 | {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, ...} | ||
2 | A000290 | {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, ...} | ||
3 | A000578 | {0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 13824, 15625, 17576, 19683, 21952, 24389, ...} | ||
4 | A000583 | {0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000, 14641, 20736, 28561, 38416, 50625, 65536, 83521, 104976, 130321, 160000, 194481, 234256, 279841, 331776, 390625, ...} | ||
5 | A000584 | {0, 1, 32, 243, 1024, 3125, 7776, 16807, 32768, 59049, 100000, 161051, 248832, 371293, 537824, 759375, 1048576, 1419857, 1889568, 2476099, 3200000, 4084101, 5153632, ...} | ||
6 | A001014 | {0, 1, 64, 729, 4096, 15625, 46656, 117649, 262144, 531441, 1000000, 1771561, 2985984, 4826809, 7529536, 11390625, 16777216, 24137569, 34012224, 47045881, 64000000, ...} | ||
7 | A001015 | {0, 1, 128, 2187, 16384, 78125, 279936, 823543, 2097152, 4782969, 10000000, 19487171, 35831808, 62748517, 105413504, 170859375, 268435456, 410338673, 612220032, ...} | ||
8 | A001016 | {0, 1, 256, 6561, 65536, 390625, 1679616, 5764801, 16777216, 43046721, 100000000, 214358881, 429981696, 815730721, 1475789056, 2562890625, 4294967296, 6975757441, ...} | ||
9 | A001017 | {0, 1, 512, 19683, 262144, 1953125, 10077696, 40353607, 134217728, 387420489, 1000000000, 2357947691, 5159780352, 10604499373, 20661046784, 38443359375, ...} | ||
10 | A008454 | {0, 1, 1024, 59049, 1048576, 9765625, 60466176, 282475249, 1073741824, 3486784401, 10000000000, 25937424601, 61917364224, 137858491849, 289254654976, ...} | ||
11 | A008455 | {0, 1, 2048, 177147, 4194304, 48828125, 362797056, 1977326743, 8589934592, 31381059609, 100000000000, 285311670611, 743008370688, 1792160394037, 4049565169664, ...} | ||
12 | A008456 | {0, 1, 4096, 531441, 16777216, 244140625, 2176782336, 13841287201, 68719476736, 282429536481, 1000000000000, 3138428376721, 8916100448256, 23298085122481, ...} |
Regular orthotopic numbers read cross-dimensionally (giving exponentials sequences)
The regular orthotopic numbers read cross-dimensionally give the exponentials sequences.
Note the disagreement about 0 0,[22] between the figurate number interpretation (which has to be 0 forn = 0 |
|
A-number |
| ||
---|---|---|---|---|
0 [22] | A000007 | {0[23], 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...} | ||
1 | A000012 | {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, 1, 1, ...} | ||
2 | A000079 | {1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, ...} | ||
3 | A000244 | {1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907, 43046721, 129140163, 387420489, 1162261467, ...} | ||
4 | A000302 | {1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144, 1048576, 4194304, 16777216, 67108864, 268435456, 1073741824, 4294967296, 17179869184, ...} | ||
5 | A000351 | {1, 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, 9765625, 48828125, 244140625, 1220703125, 6103515625, 30517578125, 152587890625, ...} | ||
6 | A000400 | {1, 6, 36, 216, 1296, 7776, 46656, 279936, 1679616, 10077696, 60466176, 362797056, 2176782336, 13060694016, 78364164096, 470184984576, ...} | ||
7 | A000420 | {1, 7, 49, 343, 2401, 16807, 117649, 823543, 5764801, 40353607, 282475249, 1977326743, 13841287201, 96889010407, 678223072849, 4747561509943, ...} | ||
8 | A001018 | {1, 8, 64, 512, 4096, 32768, 262144, 2097152, 16777216, 134217728, 1073741824, 8589934592, 68719476736, 549755813888, 4398046511104, ...} | ||
9 | A001019 | {1, 9, 81, 729, 6561, 59049, 531441, 4782969, 43046721, 387420489, 3486784401, 31381059609, 282429536481, 2541865828329, 22876792454961, ...} | ||
10 | A011557 | {1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000, 10000000000, 100000000000, 1000000000000, 10000000000000, ...} | ||
11 | A001020 | {1, 11, 121, 1331, 14641, 161051, 1771561, 19487171, 214358881, 2357947691, 25937424601, 285311670611, 3138428376721, 34522712143931, ...} | ||
12 | A001021 | {1, 12, 144, 1728, 20736, 248832, 2985984, 35831808, 429981696, 5159780352, 61917364224, 743008370688, 8916100448256, 106993205379072, ...} |
See also
Notes
- ↑ Weisstein, Eric W., Hypercube, from MathWorld—A Wolfram Web Resource.
- ↑ 2.0 2.1 Where is the
-dimensional regular convex polytope number with vertices.d - ↑ Since the power series associated with generating functions are only formal, i.e. used as placeholders for the as coefficients of , we need not worry about convergence (as long as it converges for some range of , whatever that range.)
- ↑ Herbert S. Wilf, generatingfunctionology, 2nd ed., 1994.
- ↑ Prof. Albert R. Meyer and Prof. Ronitt Rubinfeld, Generating Functions, Mathematics for Computer Science, MIT, 2005.
- ↑ Weisstein, Eric W., Fermat’s Polygonal Number Theorem, from MathWorld—A Wolfram Web Resource.
- ↑ 7.0 7.1 Weisstein, Eric W., Waring's Problem, from MathWorld—A Wolfram Web Resource.
- ↑ 8.0 8.1 8.2 8.3 Sondow, Jonathan and Weisstein, Eric W., Harmonic Number, from MathWorld—A Wolfram Web Resource. Cite error: Invalid
<ref>
tag; name "HarmonicNumber" defined multiple times with different content - ↑ 9.0 9.1 Sondow, Jonathan and Weisstein, Eric W., Riemann Zeta Function, from MathWorld—A Wolfram Web Resource.
- ↑ 10.0 10.1 Weisstein, Eric W., Schläfli Symbol, from MathWorld—A Wolfram Web Resource.
- ↑ 11.0 11.1 A057427 is the sign function ( − 1 for
, 0 forn < 0
, {{mathfont|+1} forn = 0
), while what we get here is the characteristic function of positive integers (0 forn > 0
, +1 forn ≤ 0
). Cite error: Invalidn ≥ 1 <ref>
tag; name "chi_pos_int" defined multiple times with different content - ↑ Weisstein, Eric W., Nexus Number, from MathWorld—A Wolfram Web Resource.
- ↑ 13.0 13.1 Gi-Sang Cheon and Moawwad E. A. El-Mikkawy, GENERALIZED HARMONIC NUMBER IDENTITIES AND A RELATED MATRIX REPRESENTATION, J. Korean Math. Soc. 44 (2007), No. 2, pp. 487-498.
- ↑ Downey, Lawrence M., Ong, Boon W., and Sellers, James A., Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, 2008.
- ↑ Weisstein, Eric W., Bernoulli Number, from MathWorld—A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Rising Factorial, from MathWorld—A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Multichoose, from MathWorld—A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Odd Number, from MathWorld—A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Hex Number, from MathWorld—A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Apéry's Constant, from MathWorld—A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Rhombic Dodecahedral Number, from MathWorld—A Wolfram Web Resource.
- ↑ 22.0 22.1 See 0 0 or the special case of zero to the zeroeth power.
- ↑ Note the disagreement about 0 0 between the figurate number interpretation (which has to be 0 for
) and the powers interpretation (which is 1).n = 0
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.