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Contents

Definitions

The whatever numbers are...

Formulae

The \scriptstyle n \,th whatever number is given by the formula

a(n) = ~?. \,

Recurrence relation

a(n) = k_1 ~ a(n-1) + k_2 ~ a(n-2) + k_3 ~ a(n-3),\quad n \ge 3, \,

with initial conditions

a(0) = ~?, a(1) = ~?, a(2) = ~?. \,

Generating function

G_{\{a(n)\}}(x) = ~?. \,

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 \scriptstyle k \, \scriptstyle k \,-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 \scriptstyle k \, \scriptstyle k \,-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 \scriptstyle A \, of nonnegative integers is called a basis of order \scriptstyle g \, if \scriptstyle g \, is the minimum number with the property that every nonnegative integer can be written as a sum of \scriptstyle g \, elements in \scriptstyle A \,. Lagrange’s sum of four squares can be restated as the set \scriptstyle \{ n^2 \,|\, n \,=\, 0,\, 1,\, 2,\, \ldots\} \, of nonnegative squares forms a basis of order 4.

Theorem (Cauchy) For every \scriptstyle k \,\ge\, 3 \,, the set \scriptstyle \{ P(k, n) \,|\, n \,=\, 0,\, 1,\, 2,\, \ldots \} \, of \scriptstyle k \,-gonal numbers forms a basis of order \scriptstyle k \,, i.e. every nonnegative integer can be written as a sum of \scriptstyle k \, \scriptstyle k \,-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 \scriptstyle g(d) \, such that every nonnegative integer is a sum of \scriptstyle g(d) \, \scriptstyle d \,th powers, i.e. the set \scriptstyle \{ n^d \,|\, n \,=\, 0,\, 1,\, 2,\, \ldots \} \, of \scriptstyle d \,th powers forms a basis of order \scriptstyle g(d) \,. The Hilbert-Waring problem is concerned with the study of \scriptstyle g(d) \, for \scriptstyle d \,\ge\, 2 \,. 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

a(n+1) - a(n) = ~?. \,

Partial sums

\sum_{n=1}^m a(n) = ~?, \,

where \scriptstyle t_m \, is the \scriptstyle m \,th triangular number.

Partial sums of reciprocals

\sum_{n=1}^m \frac{1}{a(n)} = ~?, \,

where \scriptstyle H_m \, is the \scriptstyle m \,th harmonic number,[2] \scriptstyle \gamma \, is the Euler-Mascheroni constant,[3] and \scriptstyle \psi(x) \, is the digamma function.[4] [5]

Sum of reciprocals

\sum_{n=1}^\infty \frac{1}{a(n)} = ~?. \,

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
N_1 \, Name Formulae

P^{(2)}_{N_1}(n) \,

n \, = 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
N_1 \, Name Generating

function

G_{P^{(2)}_{N_1}}(x) \,

Order

of basis[1]

Differences

P^{(2)}_{N_1}(n+1) - \,

P^{(2)}_{N_1}(n) \,

Partial sums

\sum_{n=1}^m P^{(2)}_{N_1}(n) \,

Partial sums of reciprocals

\sum_{n=1}^m {1\over{P^{(2)}_{N_1}(n)}} \,

Sum of Reciprocals[6][7]

\sum_{n=1}^\infty{1\over{P^{(2)}_{N_1}(n)}} \,

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 \, S_m(n),\ n \ge 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)


<!--== Wikitable color scheme is: [background: #f2f2f2; for table header] [background: #f9f9f9; for table cells] ==-->
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|+ '''Whatever numbers sequences'''
|- style="background: #f2f2f2; color: black; text-align: center;"
! width="25" style="text-align: center;" | <math>m \,</math>
! style="text-align: center;" | <math>S_m(n),\ n \ge 0 \,</math> 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?????? 
|-
|}{{nl}}

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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== See also ==

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== Notes ==
<references/>

== References ==

*
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== External links ==

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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.
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