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Regular orthotopic numbers

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The regular orthotopic numbers are a family of sequences of figurate numbers corresponding to the -dimensional regular orthotope for each dimension , where is a nonnegative integer. These include the square numbers, the cube numbers and the hypercube numbers for .

The -dimensional regular orthotopic numbers, forming regular orthotopes (e.g. points, segments, squares, cubes and then hypercubes)[1], where (-1)-cells correspond to the null polytope, 0-cells are vertices, 1-cells are edges, 2-cells are faces, and so on...
 
d = 0 Regular 0-orthotopic numbers Point numbers Form point (0 (-1)-cell facets) (regular 0-orthotope)
d = 1 Regular 1-orthotopic numbers Linear numbers Form segments (2 0-cell facets) (regular 1-orthotope)
d = 2 Regular 2-orthotopic numbers Square numbers (tetragonal numbers) Form squares (4 1-cell facets) (regular 2-orthotope)
d = 3 Regular 3-orthotopic numbers Cube numbers (hexahedral numbers) Form cubes (6 2-cell facets) (regular 3-orthotope)
d = 4 Regular 4-orthotopic numbers Octachoron numbers Form tesseracts (8 3-cell facets) (regular 4-orthotope)
d = 5 Regular 5-orthotopic numbers Decateron numbers Form penteracts (10 4-cell facets) (regular 5-orthotope)
d = 6 Regular 6-orthotopic numbers Dodecapeton numbers Form hexeracts (12 5-cell facets) (regular 6-orthotope)
d = 7 Regular 7-orthotopic numbers Tetradecahexon numbers Form hepteracts (14 6-cell facets) (regular 7-orthotope)
d = 8 Regular 8-orthotopic numbers Hexadecahepton numbers Form octeracts (16 7-cell facets) (regular 8-orthotope)
... ... ... ... ... ...
d = d Regular d-orthotopic numbers (2d) (d-1)-cell numbers Form (d)-eracts (2d (d-1)-cell facets) (regular d-orthotope)


All figurate numbers are accessible via this structured menu: Classifications of figurate numbers

Formulae

The th regular -orthotopic numbers are given by the formulae[2]

where is the dimension and is the number of nondegenerate layered regular orthotopes ( giving no dot and giving a single dot, a degenerate regular orthotope) of the -dimensional regular orthotopic number (regular -orthotope number.)

Recurrence relation

Generating function

where

is the th Eulerian polynomial (Cf. Talk:Regular_orthotopic_numbers) whose Eulerian numbers

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

[5]

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 -gonal numbers. Fermat claimed to have a proof of this result, although Fermat's proof has never been found.[6] 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 -gonal numbers (known as the polygonal number theorem,) while a vertical (higher dimensional) generalization has also been made (known as the Hilbert-Waring problem.)[7]

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 -gonal numbers forms a basis of order , i.e. every nonnegative integer can be written as a sum of -gonal 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.

The (presumed) solution to Waring's problem is (cf. A002804)

with series representation

for and .


(Presumed) solution to Waring's problem
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

[8]

Partial sums of reciprocals

[8]

Sum of reciprocals

[9]

Number of j-dimensional "vertices"

Table of formulae and values

are the number of vertices (0-dimensional elements), edges (1-dimensional elements), faces (2-dimensional elements), cells (3-dimensional elements)... respectively, where the ()-dimensional elements are the actual facets. The regular orthotopic numbers are listed by increasing number of vertices.

Regular orthotopic numbers formulae and values
d Name

Regular

-orthotope

-cell

()

Schläfli symbol[10]

Formulae


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

number

0 Point numbers

0-orthotope

zero-(-1)-cell

()

{}


0 1 1 1 1 1 1 1 1 1 1 1 1 (NOT A057427) [11]
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 1296 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


Table of related formulae and values

are the number of vertices (0-dimensional elements), edges (1-dimensional elements), faces (2-dimensional elements), cells (3-dimensional elements)... respectively, where the ()-dimensional elements are the actual facets. The regular orthotopic numbers are listed by increasing number of vertices.

Regular orthotopic numbers related formulae and values
Name

Regular -orthotope

-cell

Schläfli symbol[10]

Generating

function



where is the th Eulerian polynomial
(Cf. Talk:Regular_orthotopic_numbers)
given by
where
are the Eulerian numbers.
(Cf. triangle of Eulerian numbers)


Order

of basis

[7]

A002804

Differences[12]


Partial sums


[8][13]

Partial sums of reciprocals


[8][13]

Sum of reciprocals[14]


[9]




[15]

0 Point numbers

0-orthotope

zero-(-1)-cell

()

{}


with






1 Segment numbers

1-orthotope

di-0-cell

(2)

{}


with


[2]


 [16][17]


2 Square numbers

2-orthotope

tetra-1-cell

(4, 4)

{4}


with


Odd numbers[18]

A005408





Base 10: A013661
CFrac: A013679

3 Cube numbers

3-orthotope

hexa-2-cell

(8, 12, 6)

{4, 3}


with


Hex (or centered hexagonal)

numbers[19]

A003215


[20]


Base 10: A002117
CFrac: A013631

4 Tesseract numbers

4-orthotope

octa-3-cell

(16, 32, 24, 8)

{4, 3, 3}


with


or


Rhombic dodecahedral

numbers [21]

A005917




Base 10: A013662
CFrac: A013680

5 Penteract numbers

5-orthotope

deca-4-cell

(32, 80, 80, 40, 10)

{4, 3, 3, 3}


with





Base 10: A013663
CFrac: A013681

6 Hexeract numbers

6-orthotope

dodeca-5-cell

(64, 192, 240, 160, 60, 12)

{4, 3, 3, 3, 3}


with


or




Base 10: A013664
CFrac: A013682

7 Hepteract numbers

7-orthotope

tetradeca-6-cell

(128, 448, 672, 560, 280 , 84, 14)

{4, 3, 3, 3, 3, 3}


with





Base 10: A013665
CFrac: A013683

8 Octeract numbers

8-orthotope

hexadeca-7-cell

(256, 1024 , 1792, 1792, 1120, 448, 112, 16)

{4, 3, 3, 3, 3, 3, 3}


with


where





Base 10: A013666
CFrac: A013684

9 Enneract numbers

9-orthotope

octadeca-8-cell

(..., 18)

{4, 3, 3, 3, 3, 3, 3, 3}


with


where





Base 10: A013667
CFrac: A013685

10 Dekeract numbers

10-orthotope

icosa-9-cell

(..., 20)

{4, 3, 3, 3, 3, 3, 3, 3, 3}


with


where






Base 10: A013668
CFrac: A013686

11 Hendekeract numbers

11-orthotope

icosidi-10-cell

(..., 22)

{4, 3, 3, 3, 3, 3, 3, 3, 3, 3}


with


where





Base 10: A013669
CFrac: A013687

12 Dodekeract numbers

12-orthotope

icositetra-11-cell

(..., 24)

{4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3}


with


where






Base 10: A013670
CFrac: A013688


Table of sequences

Regular orthotopic numbers sequences
OEIS

number

sequences
0 (NOT A057427) [11] {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 for ) and the powers interpretation (which is 1.)


Regular orthotopic numbers read cross-dimensionally (giving exponentials sequences)
OEIS

number

sequences
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

  1. Weisstein, Eric W., Hypercube, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/Hypercube.html]
  2. 2.0 2.1 Where is the -dimensional regular convex polytope number with vertices.
  3. 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.)
  4. Herbert S. Wilf, generatingfunctionology, 2nd ed., 1994.
  5. Prof. Albert R. Meyer and Prof. Ronitt Rubinfeld, Generating Functions, Mathematics for Computer Science, MIT, 2005.
  6. Weisstein, Eric W., Fermat's Polygonal Number Theorem, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/FermatsPolygonalNumberTheorem.html]
  7. 7.0 7.1 Weisstein, Eric W., Waring's Problem, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/WaringsProblem.html]
  8. 8.0 8.1 8.2 8.3 Sondow, Jonathan and Weisstein, Eric W., Harmonic Number, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/HarmonicNumber.html] Cite error: Invalid <ref> tag; name "HarmonicNumber" defined multiple times with different content
  9. 9.0 9.1 Sondow, Jonathan and Weisstein, Eric W., Riemann Zeta Function, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/RiemannZetaFunction.html]
  10. 10.0 10.1 Weisstein, Eric W., Schläfli Symbol, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/SchlaefliSymbol.html]
  11. 11.0 11.1 A057427 is the sign function (-1 for n < 0, 0 for n = 0, +1 for n > 0,) while what we get here is the characteristic function of positive integers (0 for n ≤ 0, +1 for n ≥ 1.)
  12. Weisstein, Eric W., Nexus Number, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/NexusNumber.html]
  13. 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.
  14. Downey, Lawrence M., Ong, Boon W., and Sellers, James A., Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, 2008.
  15. Weisstein, Eric W., Bernoulli Number, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/BernoulliNumber.html]
  16. Weisstein, Eric W., Rising Factorial, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/RisingFactorial.html]
  17. Weisstein, Eric W., Multichoose, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/Multichoose.html]
  18. Weisstein, Eric W., Odd Number, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/OddNumber.html]
  19. Weisstein, Eric W., Hex Number, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/HexNumber.html]
  20. Weisstein, Eric W., Apéry's Constant, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/Ap%C3%A9rysConstant.html]
  21. Weisstein, Eric W., Rhombic Dodecahedral Number, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/RhombicDodecahedralNumber.html]
  22. 22.0 22.1 Cf. 0^0 or the special case of zero to the zeroeth power.
  23. Note the disagreement about 0^0 between the figurate number interpretation (which has to be 0 for ) and the powers interpretation (which is 1.)

External links