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Definitions

The whatever numbers are...

Formulae

The th whatever number is given by the formula

Recurrence relation

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 -gonal numbers. Fermat claimed to have a proof of this result, although Fermat's proof has never been found.[1] 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.)

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 the last 90 years, and it is still a very active area of research.

Forward differences

Partial sums

where is the th triangular number.

Partial sums of reciprocals

where is the th harmonic number,[2] is the Euler-Mascheroni constant,[3] and is the digamma function.[4] [5]

Sum of reciprocals

Tables

Table of formulae and values

Polygonal numbers associated with constructible polygons (with straightedge and compass) (Cf. A003401) are named in bold.

Polygonal numbers formulae and values
Name Formulae

= 0 1 2 3 4 5 6 7 8 9 10 11 12 A-number
3 Triangular numbers A??????
4 Square numbers A??????
5 Pentagonal numbers A??????
6 Hexagonal numbers A??????
7 Heptagonal numbers A??????
8 Octagonal numbers A??????
9 Nonagonal numbers A??????
10 Decagonal numbers A??????
11 Hendecagonal numbers A??????
12 Dodecagonal numbers A??????
13 Tridecagonal numbers A??????
14 Tetradecagonal numbers A??????
15 Pentadecagonal numbers A??????
16 Hexadecagonal numbers A??????
17 Heptadecagonal numbers A??????
18 Octadecagonal numbers A??????
19 Nonadecagonal numbers A??????
20 Icosagonal numbers A??????
21 Icosihenagonal numbers A??????
22 Icosidigonal numbers A??????
23 Icositrigonal numbers A??????
24 Icositetragonal numbers A??????
25 Icosipentagonal numbers A??????
26 Icosihexagonal numbers A??????
27 Icosiheptagonal numbers A??????
28 Icosioctagonal numbers A??????
29 Icosinonagonal numbers A??????
30 Triacontagonal numbers A??????

Table of related formulae and values

Polygonal numbers associated with constructible polygons (with straightedge and compass) (Cf. A003401) are named in bold.

Polygonal numbers related formulae and values
Name Generating

function

Order

of basis[1]

Differences

Partial sums

Partial sums of reciprocals

Sum of Reciprocals[6][7]

3 Triangular numbers
4 Square numbers
5 Pentagonal numbers
6 Hexagonal numbers
7 Heptagonal numbers
8 Octagonal numbers
9 Nonagonal numbers
10 Decagonal numbers
11 Hendecagonal numbers
12 Dodecagonal numbers
13 Tridecagonal numbers
14 Tetradecagonal numbers
15 Pentadecagonal numbers
16 Hexadecagonal numbers
17 Heptadecagonal numbers
18 Octadecagonal numbers
19 Nonadecagonal numbers
20 Icosagonal numbers
21 Icosihenagonal numbers
22 Icosidigonal numbers
23 Icositrigonal numbers
24 Icositetragonal numbers
25 Icosipentagonal numbers
26 Icosihexagonal numbers
27 Icosiheptagonal numbers
28 Icosioctagonal numbers
29 Icosinonagonal numbers
30 Triacontagonal numbers

Table of sequences

Whatever numbers sequences
m
Sm(n), n   ≥   0,
sequences
A-number
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, 0, 0, 0, 0, 0, ...} A??????
1 {0, ...} A??????
2 {0, ...} A??????
3 {0, ...} A??????
4 {0, ...} A??????
5 {0, ...} A??????
6 {0, ...} A??????
7 {0, ...} A??????
8 {0, ...} A??????
9 {0, ...} A??????
10 {0, ...} A??????
11 {0, ...} A??????
12 {0, ...} A??????
13 {0, ...} A??????
14 {0, ...} A??????
15 {0, ...} A??????
16 {0, ...} A??????
17 {0, ...} A??????
18 {0, ...} A??????
19 {0, ...} A??????
20 {0, ...} A??????
21 {0, ...} A??????
22 {0, ...} A??????
23 {0, ...} A??????
24 {0, ...} A??????
25 {0, ...} A??????
26 {0, ...} A??????
27 {0, ...} A??????
28 {0, ...} A??????
29 {0, ...} A??????
30 {0, ...} A??????

Table of sequences (copy and paste)

{| class="wikitable" cellspacing="0" cellpadding="4px" style="margin: 1em auto; border-collapse: collapse; border: 1px solid darkgray; background: #f9f9f9; color: black; empty-cells: show; text-align: left;"
|+ '''Whatever numbers sequences'''
|- style="background: #f2f2f2; color: black; text-align: center;"
! width="25" style="text-align: center;" | {{math|''m''|tex = m|&}}
! style="text-align: center;" | {{math|''S''{{sub|''m''}}(''n''), ''n'' {{rel|ge}} 0,|tex = S_m(n), n \ge 0,|&}} sequences
! width="75" style="text-align: center;" | [[A-number]]
|-
| style="text-align: center;" | '''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, 0, 0, 0, 0, 0, ...}
| style="text-align: center;" | A??????
|-
| style="text-align: center;" | '''1'''
| {0, ...} 
| style="text-align: center;" | A??????
|-
| style="text-align: center;" | '''2'''
| {0, ...} 
| style="text-align: center;" | A??????
|-
| style="text-align: center;" | '''3'''
| {0, ...} 
| style="text-align: center;" | A??????
|-
| style="text-align: center;" | '''4'''
| {0, ...} 
| style="text-align: center;" | A??????
|-
| style="text-align: center;" | '''5'''
| {0, ...}
| style="text-align: center;" | A??????
|-
| style="text-align: center;" | '''6'''
| {0, ...}
| style="text-align: center;" | A??????
|-
| style="text-align: center;" | '''7'''
| {0, ...}
| style="text-align: center;" | A??????
|-
| style="text-align: center;" | '''8'''
| {0, ...}
| style="text-align: center;" | A??????
|-
| style="text-align: center;" | '''9'''
| {0, ...}
| style="text-align: center;" | A??????
|-
| style="text-align: center;" | '''10'''
| {0, ...}
| style="text-align: center;" | A?????? 
|-
| style="text-align: center;" | '''11'''
| {0, ...}
| style="text-align: center;" | A??????
|-
| style="text-align: center;" | '''12'''
| {0, ...}
| style="text-align: center;" | A??????
|-
| style="text-align: center;" | '''13'''
| {0, ...}
| style="text-align: center;" | A??????
|-
| style="text-align: center;" | '''14'''
| {0, ...}
| style="text-align: center;" | A?????? 
|-
| style="text-align: center;" | '''15'''
| {0, ...}
| style="text-align: center;" | A??????
|-
| style="text-align: center;" | '''16'''
| {0, ...}
| style="text-align: center;" | A?????? 
|-
| style="text-align: center;" | '''17'''
| {0, ...}
| style="text-align: center;" | A??????
|-
| style="text-align: center;" | '''18'''
| {0, ...}
| style="text-align: center;" | A??????
|-
| style="text-align: center;" | '''19'''
| {0, ...}
| style="text-align: center;" | A??????
|-
| style="text-align: center;" | '''20'''
| {0, ...}
| style="text-align: center;" | A??????
|-
| style="text-align: center;" | '''21'''
| {0, ...}
| style="text-align: center;" | A??????
|-
| style="text-align: center;" | '''22'''
| {0, ...}
| style="text-align: center;" | A??????
|-
| style="text-align: center;" | '''23'''
| {0, ...}
| style="text-align: center;" | A?????? 
|-
| style="text-align: center;" | '''24'''
| {0, ...}
| style="text-align: center;" | A??????
|-
| style="text-align: center;" | '''25'''
| {0, ...}
| style="text-align: center;" | A??????
|-
| style="text-align: center;" | '''26'''
| {0, ...}
| style="text-align: center;" | A??????
|-
| style="text-align: center;" | '''27'''
| {0, ...}
| style="text-align: center;" | A??????
|-
| style="text-align: center;" | '''28'''
| {0, ...}
| style="text-align: center;" | A??????
|-
| style="text-align: center;" | '''29'''
| {0, ...}
| style="text-align: center;" | A??????
|-
| style="text-align: center;" | '''30'''
| {0, ...}
| style="text-align: center;" | A?????? 
|-
|}

Number triangles

Rectangular number triangle (n = [0..12], d = [0..12])

Cf. User:Daniel Forgues/Copy and paste/Rectangular number triangle.

Equilateral number triangle (n = [1..15], i = [1..15])

Cf. User:Daniel Forgues/Copy and paste/Equilateral number triangle

Sections (copy and paste)


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Notes

  1. 1.0 1.1 Weisstein, Eric W., Fermat's Polygonal Number Theorem, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/FermatsPolygonalNumberTheorem.html]
  2. Sondow, Jonathan and Weisstein, Eric W., Harmonic Number, From MathWorld--A Wolfram Web Resource.
  3. Weisstein, Eric W., Euler-Mascheroni Constant, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/Euler-MascheroniConstant.html]
  4. Weisstein, Eric W., Digamma Function, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/DigammaFunction.html]
  5. Weisstein, Eric W., Polygamma Function, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/PolygammaFunction.html]
  6. Downey, Lawrence M., Ong, Boon W., and Sellers, James A., Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, 2008.
  7. PSYCHEDELIC GEOMETRY, INVERSE POLYGONAL NUMBERS SERIES.