Polygonal numbers

Triangular numbers	Square numbers	Pentagonal numbers	Hexagonal numbers

The polygonal numbers are the family of sequences of 2-dimensional convex regular polytope numbers, made of

n

successive polygonal layers with a constant number

N0

of 0-dimensional elements (vertices

V

of the polygons), having

n

dots for each edge (including both end vertices) of the

n

th layer,

n   ≥   1

, with all layers sharing a common vertex (which corresponds to

n = 1

) and having two sides sharing that vertex. The number

N1

of 1-dimensional elements (edges

E

of the polygons) equals the number

N0

of 0-dimensional elements (vertices

V

of the polygons).

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

Formulae

The

n

th

N0

-gonal number is given by the formulae

P   (2) N0 (n) :=   n ∑   i   = 0    P   (1) N0 −1 (i) = n + (N0 − 2) P   (2) 3 (n − 1) = n + (N0 − 2) Tn  − 1 = n + (N0 − 2) (  n2  )  = n + (N0 − 2) (n − 1) n 2 = n 2 [(N0 − 2) n − (N0 − 4)],

where

P   (1) N0 (n)

is the

n

th

N0

-gonal gnomonic number, and where

N0

is the number of 0-dimensional elements (which are vertices

V

 ) of the polygons and

Tn

is the

n

th triangular number.

Nontrivial polygonal numbers

A number which is non-trivially polygonal (a nontrivial polygonal number?) is a number

n

which is congruent to

k  (mod tk  − 1)

with

3   ≤   tk  − 1   <   n

, i.e.

n = j ⋅  tk  − 1 + k

with

k   ≥   3

and

j   ≥   1

, is an

m

-gonal number of order

k

, with

m = j + 2

. This number is composite, with nontrivial factorization

n = k 2 ( j​k  −  j + 2)

.
For example, for

n = 45

, we have

• 45 mod t3  − 1 = 45 mod 3 = 3

  , thus of order

  k = 3

  ;

• 45 mod t4  − 1 = 45 mod 6 = 3

  ;

• 45 mod t5  − 1 = 45 mod 10 = 5

  , thus of order

  k = 5

  ;

• 45 mod t6  − 1 = 45 mod 15 = 0

  ;

• 45 mod t7  − 1 = 45 mod 21 = 3

  ;

• 45 mod t8  − 1 = 45 mod 28 = 17

  ;

• 45 mod t9  − 1 = 45 mod 36 = 9

  , thus of order

  k = 9

  ;

where

m = 2 n + 2 k (k  −  2) k (k  −  1) = n + k (k  −  2) tk  − 1

yields

m = 16, 6, 3,

for orders

k = 3, 5, 9,

respectively.

m

-gonal numbers

n

for

3   ≤   m   <   n

(    : units;     : primes;     : missed composites)

m = 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 Y 7   8   9   Y 10 Y   11     12     Y 13       14       15 Y     Y 16   Y     17         18         Y 19           20           21 Y         Y 22     Y       23             24             Y 25   Y           26               27               Y 28 Y     Y         29                 30                 Y 31                   32

m = 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 33                   Y 34         Y           35     Y               36 Y Y                 Y 37                       38                       39                       Y 40           Y             41                         42                         Y 43                           44                           45 Y     Y                   Y 46             Y               47                             48                             Y 49   Y                           50                               51     Y                         Y 52               Y                 53                                 54                                 Y 55 Y       Y                         56                                   57                                   Y 58                 Y                   59                                     60                                     Y 61                                       62                                       63                                       Y 64   Y               Y

Composite numbers n which are not m-gonal numbers for 3   ≤   m < n

In the following, for

n   ≥   3

,

n

is trivially an

n

-gonal number, of order

k = 2

. So,

n

is a nontrivial polygonal number means that it is also an

m

-gonal number with

m < n

, hence of order

k > 2

.
For [composite] positive integers

n

which are nontrivial polygonal numbers, i.e.

n = P  (2) m (k)

for some order

k   ≥   3

, we get the nontrivial factorization

n = k 2 [(m  −  2) k  −  (m  −  4)]

. Note that while nontrivial polygonal numbers are necessarily composite, unfortunately the converse is not true: not all composite numbers are nontrivial polygonal numbers.
Since we are looking for solutions of

(m  −  2) k 2  −  (m  −  4) k  −  2 n = 0

, with

m   ≥   3

and

k   ≥   3

, the largest order

k

we need to consider is

k = (m  −  4) + 2√  (m  −  4) 2 + 8 (m  −  2) n 2 (m  −  2)

with

m = 3

, thus

3 ≤ k ≤ −1 + 2√  1 + 8 n 2 .

Or, since we are looking for solutions of

2 n = m k  (k  −  1)  −  2 k  (k  −  2)

, with

m   ≥   3

and

k   ≥   3

, the largest

m

we need to consider is

m = 2 n + 2 k  (k  −  2) k  (k  −  1)

with order

k = 3

, thus

3 ≤ m ≤ n + 3 3 .

A176949 Composite numbers

n

for which A176948

(n) = n

. (Composite numbers

n

which are not

m

-gonal number for

3   ≤   m   <   n

.)

{4, 8, 14, 20, 26, 32, 38, 44, 50, 56, 62, 68, 74, 77, 80, 86, 98, 104, 110, 116, 119, 122, 128, 134, 140, 143, 146, 152, 158, 161, 164, 170, 182, 187, 188, 194, 200, 203, 206, 209, 212, 218, 221, 224, 230, 236, 242, 248, 254, 266, 272, 278, 284, 290, 296, 299, 302, ...}

Even composite numbers n which are not m-gonal numbers for 3   ≤   m < n

The even numbers

n   ≥   4

which are not

m

-gonal number for

3   ≤   m   <   n

are all coprime to 3, since composite numbers

n

which are divisible by 3 are

m

-gonal numbers of order

k = 3

, with

m = n + 3 3

.
The even composite numbers

n   ≥   10

which are congruent to 4  (mod 6), i.e.

n = 6 j + 4

for

j   ≥   1

are

m

-gonal numbers of order

k = 4

, with

m = j + 2

. Thus, the even numbers

n   ≥   4

which are not

m

-gonal number for

3   ≤   m   <   n

, with the exception of 4 (the only square number, 4-gonal of rank 2), are all congruent to 2  (mod 6) although some numbers don’t show up: from 8 to 302, the numbers 92, 176 and 260 are missing (since they are nontrivial polygonal numbers).
A274968 Even composite numbers

n

for which A176948

(n) = n

. (Even numbers

n   ≥   4

which are not

m

-gonal number for

3   ≤   m   <   n

.)

{4, 8, 14, 20, 26, 32, 38, 44, 50, 56, 62, 68, 74, 80, 86, 98, 104, 110, 116, 122, 128, 134, 140, 146, 152, 158, 164, 170, 182, 188, 194, 200, 206, 212, 218, 224, 230, 236, 242, 248, 254, 266, 272, 278, 284, 290, 296, 302, ...}

Odd composite numbers n which are not m-gonal numbers for 3   ≤   m < n

The odd composite numbers

n

which are not

m

-gonal number for

3   ≤   m   <   n

are all coprime to 30, since

• composite numbers

  n

  which are divisible by 3 are

  m

  -gonal numbers of order 3, with

  m = n + 3 3

  ;
• odd composite numbers

  n

  which are divisible by 5 are

  m

  -gonal numbers of order 5, with

  m = n + 15 10

  .

A274967 Odd composite numbers

n

for which A176948

(n) = n

. (Odd composite numbers

n

which are not

m

-gonal number for

3   ≤   m   <   n

.)

{77, 119, 143, 161, 187, 203, 209, 221, 299, 319, 323, 329, 371, 377, 391, 407, 413, 437, 473, 493, 497, 517, 527, 533, 539, 551, 581, 583, 589, 611, 623, 629, 649, 667, 689, 707, 713, 731, 737, 749, 767, 779, 791, 799, 803, 817, 851, 869, 893, 899, 901, 913, ...}

Sequences

A177025 Number of ways to represent 

n   ≥   3

as a polygonal number.

{1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 3, 2, 1, 2, 1, 1, 3, 2, 1, 2, 2, 1, 2, 3, 1, 2, 1, 1, 2, 2, 2, 4, 1, 1, 2, 2, 1, 2, 1, 1, 4, 2, 1, 2, 2, 1, 3, 2, 1, 2, 3, 1, 2, 2, 1, 2, 1, 1, 2, 3, 2, 4, 1, 1, 2, 3, 1, 2, 1, 1, 3, 2, 1, 3, 1, 1, 4, 2, 1, 2, 2, 1, 2, 2, 1, 2, 3, 2, 2, 2, 2, 3, 1, 1, 2, 3, ...}

A063778

a (n)

is the least integer that is polygonal in exactly 

n

ways.

{3, 6, 15, 36, 225, 561, 1225, 11935, 11781, 27405, 220780, 203841, 3368925, 4921840, 7316001, 33631521, 142629201, 879207616, 1383958576, 3800798001, 12524486976, 181285005825, 118037679760, 239764947345, 738541591425, 1289707733601, 1559439365121, ...}

A176774 Smallest polygonality of 

n

= smallest integer 

m   ≥   3

such that 

n, n   ≥   3,

is 

m

-gonal number.

{3, 4, 5, 3, 7, 8, 4, 3, 11, 5, 13, 14, 3, 4, 17, 7, 19, 20, 3, 5, 23, 9, 4, 26, 10, 3, 29, 11, 31, 32, 12, 7, 5, 3, 37, 38, 14, 8, 41, 15, 43, 44, 3, 9, 47, 17, 4, 50, 5, 10, 53, 19, 3, 56, 20, 11, 59, 21, 61, 62, 22, 4, 8, 3, 67, 68, 24, 5, 71, 25, 73, 74, 9, 14, 77, 3, 79, 80, 4, 15, 83, ...}

A176948

a (n)

is the smallest solution

x

to A176774

(x) = n, n   ≥   3; a (n) = 0

if this equation has no solution.

{3, 4, 5, 0, 7, 8, 24, 27, 11, 33, 13, 14, 42, 88, 17, 165, 19, 20, 60, 63, 23, 69, 72, 26, 255, 160, 29, 87, 31, 32, 315, 99, 102, 208, 37, 38, 114, 805, 41, 123, 43, 44, 132, 268, 47, 696, 475, 50, 150, 304, 53, 159, 162, 56, 168, 340, 59, 177, 61, 62, 615, 1309, 192, 388, ...}

A176775 Index of

n, n   ≥   3,

as

m

-gonal number for the smallest possible

m

(= A176774

(n)

).

{2, 2, 2, 3, 2, 2, 3, 4, 2, 3, 2, 2, 5, 4, 2, 3, 2, 2, 6, 4, 2, 3, 5, 2, 3, 7, 2, 3, 2, 2, 3, 4, 5, 8, 2, 2, 3, 4, 2, 3, 2, 2, 9, 4, 2, 3, 7, 2, 6, 4, 2, 3, 10, 2, 3, 4, 2, 3, 2, 2, 3, 8, 5, 11, 2, 2, 3, 7, 2, 3, 2, 2, 5, 4, 2, 12, 2, 2, 9, 4, 2, 3, 5, 2, 3, 4, 2, 3, 13, 8, 3, 4, 5, 6, 2, 2, 3, 10, 2, 3, 2, 2, ...}

A090466 Regular figurative or polygonal numbers of order greater than 2. (Composite numbers which are nontrivial polygonal numbers.)

{6, 9, 10, 12, 15, 16, 18, 21, 22, 24, 25, 27, 28, 30, 33, 34, 35, 36, 39, 40, 42, 45, 46, 48, 49, 51, 52, 54, 55, 57, 58, 60, 63, 64, 65, 66, 69, 70, 72, 75, 76, 78, 81, 82, 84, 85, 87, 88, 90, 91, 92, 93, 94, 95, 96, 99, 100, 102, 105, 106, 108, 111, 112, 114, 115, 117, 118, ...}

A090467 Numbers which are not regular figurative or polygonal numbers of order greater than 2. That is, numbers not of the form

1 + k  n  (n  −  1) 2  −  (n  −  1) 2

, where

n   ≥   2

and

k   ≥   2

. (Union of unit 1, the primes and composite numbers which are NOT nontrivial polygonal numbers.)

{1, 2, 3, 4, 5, 7, 8, 11, 13, 14, 17, 19, 20, 23, 26, 29, 31, 32, 37, 38, 41, 43, 44, 47, 50, 53, 56, 59, 61, 62, 67, 68, 71, 73, 74, 77, 79, 80, 83, 86, 89, 97, 98, 101, 103, 104, 107, 109, 110, 113, 116, 119, 122, 127, 128, 131, 134, 137, 139, 140, 143, 146, 149, 151, 152, ...}

Polygonal roots

The

k

-gonal roots

rk

of

n

are defined as the roots of the quadratic equation

${\begin{array}{l}\displaystyle {n={\frac {r_{k}}{2}}\,\left[(k-2)\,r_{k}-(k-4)\right],}\end{array}}$

hence

${\begin{array}{l}\displaystyle {(k-2)\,{r_{k}}^{2}-(k-4)r_{k}-2n=0,}\end{array}}$

yielding the quadratic roots

${\begin{array}{l}\displaystyle {r_{k}={\frac {(k-4)\pm {\sqrt {(k-4)^{2}+8n\,(k-2)}}}{2\,(k-2)}}.}\end{array}}$

For example, the trigonal roots (i.e. triangular roots) of

n

are

${\begin{array}{l}\displaystyle {r_{3}={\frac {-1\pm {\sqrt {1+8n}}}{2}},}\end{array}}$

the tetragonal roots (i.e. square roots) of

n

are

${\begin{array}{l}\displaystyle {r_{4}=\pm {\sqrt {n}},}\end{array}}$

and the pentagonal roots of

n

are

${\begin{array}{l}\displaystyle {r_{5}={\frac {1\pm {\sqrt {1+24n}}}{6}}.}\end{array}}$

The above definition of

k

-gonal roots applies for any

n ∈ ℂ

. One can verify that the triangular roots of 10 are

r3 (10) =  − 1  ±  9 2 = { −  5, 4}

and the pentagonal roots of 12 are

r5 (12) = 1  ±  17 6 = { −  8 3 , 3}

.

Oblong roots

Since oblong numbers are twice the triangular numbers, we have (by replacing

2 n

by

n

in the above triangular roots formula)

${\begin{array}{l}\displaystyle {n=r\,\left[(k-2)\,r-(k-4)\right],\quad k=3,}\end{array}}$

i.e.

${\begin{array}{l}\displaystyle {n=r\,(r+1),}\end{array}}$

yielding the oblong roots

${\begin{array}{l}\displaystyle {r={\frac {-1\pm {\sqrt {1+4n}}}{2}}.}\end{array}}$

One can verify that the oblong roots of 30 are

r =  − 1  ±  11 2 = { − 6, 5}

.

Schläfli–Poincaré (convex) polytope formula

Schläfli–Poincaré generalization of the Descartes–Euler (convex) polyhedral formula.

For nondegenerate 2-dimensional regular convex polygons:

$\sum _{i=0}^{1}(-1)^{i}N_{i}=N_{0}-N_{1}=V-E=0,\,$

where

N0

is the number of 0-dimensional elements (vertices

V

 ),

N1

is the number of 1-dimensional elements (edges

E

 ) of the convex polygon.

Recurrence equation

$P_{N_{0}}^{(2)}(n)=3P_{N_{0}}^{(2)}(n-1)-3P_{N_{0}}^{(2)}(n-2)+P_{N_{0}}^{(2)}(n-3),\quad n>2,\,$

with initial conditions

$P_{N_{0}}^{(2)}(0)=0,\ P_{N_{0}}^{(2)}(1)=1,\ P_{N_{0}}^{(2)}(2)=N_{0}.\,$

Generating function

$G_{\{P_{N_{0}}^{(2)}(n)\}}(x)={\frac {x\,\left((N_{0}-3)\,x+1\right)}{(1-x)^{3}}}.\,$

Order of basis

The order of basis of

N0

-gonal numbers is:

$g_{\{P_{N_{0}}^{(2)}\}}=N_{0},\quad N_{0}\geq 3.\,$

The order of basis

g

for numbers of the form

k n  +  1, k > 0,

is

k

, since to represent the numbers in the congruence classes

{0, 1, …, k  −  1}

by adding numbers congruent to

1  (mod k)

we need as many terms as the class number, for each congruence classes, e.g. for

k = 5

:

numbers of form

5 n + 1

are expressible as 1 term of the form

5n + 1

;

numbers of form

5 n + 2

are expressible as the sum of 2 terms of the form

5 n + 1

;

numbers of form

5 n + 3

are expressible as the sum of 3 terms of the form

5 n + 1

;

numbers of form

5 n + 4

are expressible as the sum of 4 terms of the form

5 n + 1

;

numbers of form

5 n + 0

are expressible as the sum of 5 terms of the form

5 n + 1

.

In 1638, Fermat proposed that every positive integer is a sum of at most three triangular numbers, four square numbers, five pentagonal numbers, and

k

k

-gonal numbers. Fermat claimed to have a proof of this result, although Fermat’s proof has never been found. 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

k

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

A

of nonnegative integers is called a basis of order

g

if

g

is the minimum number with the property that every nonnegative integer can be written as a sum of

g

elements in

A

. Lagrange’s sum of four squares can be restated as the set

{n 2 | n = 0, 1, 2, …}

of nonnegative squares forms a basis of order 4. Theorem (Cauchy) For every

k   ≥   3

, the set

{P (k, n) | n = 0, 1, 2, …}

of

k

-gon numbers forms a basis of order

k

, i.e. every nonnegative integer can be written as a sum of

k

k

-gon 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

g (d)

such that every nonnegative integer is a sum of

g (d)

d

 th powers, i.e. the set

{n d | n = 0, 1, 2, …}

of

d

 th powers forms a basis of order

g (d)

. The Hilbert–Waring problem is concerned with the study of

g (d)

for

d   ≥   2

. 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.

In 1997, Conway et al. proved a theorem, called the fifteen theorem, which states that, if a positive definite quadratic form with integer matrix entries represents all natural numbers up to 15, then it represents all natural numbers. This theorem contains Lagrange’s four-square theorem, since every number up to 15 is the sum of at most four squares.

Differences

$P_{N_{0}}^{(2)}(n)-P_{N_{0}}^{(2)}(n-1)=P_{N_{0}-1}^{(1)}(n)=(N_{0}-2)(n-1)+1,\,$

where

P (1) N0  −  1 (n)

is the

n

th

N0

-gonal gnomonic number.

Partial sums

$\sum _{n=1}^{m}P_{N_{0}}^{(2)}(n)=Y_{N_{0}+1}^{(3)}(m)={\frac {1}{6}}m(m+1)[(N_{0}-2)m-(N_{0}-5)]={\frac {1}{3}}P_{3}^{(2)}(m)[(N_{0}-2)m-(N_{0}-5)]={\frac {1}{3}}T_{m}[(N_{0}-2)m-(N_{0}-5)],\,$

where

Tm

is the

m

th triangular number and

Y  (3) N0  + 1 (m)

is the

m

th

N0

-gonal pyramidal number.

Partial sums of reciprocals

For

N0   ≠   4

,

${\begin{array}{l}\displaystyle {\begin{aligned}\sum _{n=1}^{m}{\frac {1}{P_{N_{0}}^{(2)}(n)}}&=-{\frac {2\left(H_{m}-\psi (m+{\frac {2}{N_{0}-2}})+\psi ({\frac {2}{N_{0}-2}})\right)}{(N_{0}-4)}}\\&=-{\frac {2\left(\psi (m+1)+\gamma -\psi (m+{\frac {2}{N_{0}-2}})+\psi ({\frac {2}{N_{0}-2}})\right)}{(N_{0}-4)}}\\&={\frac {2\left(\psi (m+{\frac {2}{N_{0}-2}})-\psi (m+1)-\psi ({\frac {2}{N_{0}-2}})-\gamma \right)}{(N_{0}-4)}},\end{aligned}}\end{array}}$

where

Hm

is the

m

th harmonic number,

γ

is the Euler–Mascheroni constant, and

ψ(x)

is the digamma function.
For

N0 = 4

,

$\sum _{n=1}^{m}{\frac {1}{P_{4}^{(2)}(n)}}=H_{m}^{(2)}.\,$

Sum of reciprocals

For

N0   ≠   4

,

$\sum _{n=1}^{\infty }{\frac {1}{P_{N_{0}}^{(2)}(n)}}=-{\frac {2\left(\psi \left({\frac {2}{N_{0}-2}}\right)+\gamma \right)}{(N_{0}-4)}}.\,$

For

N0 = 4

, the sum of reciprocals of the square numbers

$\sum _{n=1}^{\infty }{\frac {1}{P_{4}^{(2)}(n)}}=\zeta (2)\,$

can be interpreted as

1 p

, where

p = 1 ζ (2)

is the probability that a random integer

x

is squarefree or that two random integers

x

and

y

are coprime, i.e. the random integer

x y

is squarefree.

Table of formulae and values

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

Polygonal numbers formulae and values

N0	Name	P   (2) N0(n) = n 2 [(N0  −  2) n  −  (N0  −  4)]	0	1	2	3	4	5	6	7	8	9	10	11	12	A-numbers
3	Triangular	n (n + 1) / 2	0	1	3	6	10	15	21	28	36	45	55	66	78	A000217
4	Square	n 2 P   (2) 3(n  −  1) + P   (2) 3(n) n + 2 P   (2) 3(n  −  1)	0	1	4	9	16	25	36	49	64	81	100	121	144	A000290
5	Pentagonal	n  (3n  −  1) / 2	0	1	5	12	22	35	51	70	92	117	145	176	210	A000326
6	Hexagonal	n  (2n  −  1)	0	1	6	15	28	45	66	91	120	153	190	231	276	A000384
7	Heptagonal	n  (5n  −  3) / 2	0	1	7	18	34	55	81	112	148	189	235	286	342	A000566
8	Octagonal	n  (3n  −  2)	0	1	8	21	40	65	96	133	176	225	280	341	408	A000567
9	9-gonal	n  (7n  −  5) / 2	0	1	9	24	46	75	111	154	204	261	325	396	474	A001106
10	10-gonal	n  (4n  −  3)	0	1	10	27	52	85	126	175	232	297	370	451	540	A001107
11	11-gonal	n  (9n  −  7) / 2	0	1	11	30	58	95	141	196	260	333	415	506	606	A051682
12	12-gonal	n  (5n  −  4)	0	1	12	33	64	105	156	217	288	369	460	561	672	A051624
13	13-gonal	n  (11n  −  9) / 2	0	1	13	36	70	115	171	238	316	405	505	616	738	A051865
14	14-gonal	n  (6n  −  5)	0	1	14	39	76	125	186	259	344	441	550	671	804	A051866
15	15-gonal	n  (13n  −  11) / 2	0	1	15	42	82	135	201	280	372	477	595	726	870	A051867
16	16-gonal	n  (7n  −  6)	0	1	16	45	88	145	216	301	400	513	640	781	936	A051868
17	17-gonal	n  (15n  −  13) / 2	0	1	17	48	94	155	231	322	428	549	685	836	1002	A051869
18	18-gonal	n  (8n  −  7)	0	1	18	51	100	165	246	343	456	585	730	891	1068	A051870
19	19-gonal	n  (17n  −  15) / 2	0	1	19	54	106	175	261	364	484	621	775	946	1134	A051871
20	20-gonal	n  (9n  −  8)	0	1	20	57	112	185	276	385	512	657	820	1001	1200	A051872
21	21-gonal	n  (19n  −  17) / 2	0	1	21	60	118	195	291	406	540	693	865	1056	1266	A051873
22	22-gonal	n  (10n  −  9)	0	1	22	63	124	205	306	427	568	729	910	1111	1332	A051874
23	23-gonal	n  (21n  −  19) / 2	0	1	23	66	130	215	321	448	596	765	955	1166	1398	A051875
24	24-gonal	n  (11n  −  10)	0	1	24	69	136	225	336	469	624	801	1000	1221	1464	A051876
25	25-gonal	n  (23n  −  21) / 2	0	1	25	72	142	235	351	490	652	837	1045	1276	1530	A255184
26	26-gonal	n  (12n  −  11)	0	1	26	75	148	245	366	511	680	873	1090	1331	1596	A255185
27	27-gonal	n  (25n  −  23) / 2	0	1	27	78	154	255	381	532	708	909	1135	1386	1662	A255186
28	28-gonal	n  (13n  −  12)	0	1	28	81	160	265	396	553	736	945	1180	1441	1728	A161935  (n  −  1), n   ≥   1
29	29-gonal	n  (27n  −  25) / 2	0	1	29	84	166	275	411	574	764	981	1225	1496	1794	A255187
30	30-gonal	n  (14n  −  13)	0	1	30	87	172	285	426	595	792	1017	1270	1551	1860	A254474

Table of related formulae and values

N0

and

N1

are the number of vertices (0-dimensional) and edges (1-dimensional) respectively, where the edges are the actual facets. The regular Platonic numbers are listed by increasing number

N0

of vertices, which equals the number

N1

of facets, or sides of the polygons.

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

Polygonal numbers related formulae and values

N0	Name (N0, N1) Schläfli symbol	Generating function G{P   (2) N0(n)}(x) = x [(N0  −  3) x + 1] (1  −  x) 3	Order of basis g{P   (2) N0} N0, N0   ≥   3	Differences Gnomonic numbers P   (2) N0(n)  −  P   (2) N0(n  −  1) = P   (1) N0 − 1(n) = (N0  −  2) (n  −  1) + 1	Partial sums   m ∑   n   = 1    P   (2) N0(n) = Y  (3) N0 + 1(m) = Tm 3 [(N0  −  2) m  −  (N0  −  5)]	Partial sums of reciprocals   m ∑   n   = 1    1 P   (2) N0(n) = 2 ψ m + 2 N0  −  2  −  ψ (m + 1)  −  ψ  2 N0  −  2  −  γ (N0  −  4) , N0   ≠   4.	Sum of Reciprocals   ∞ ∑   n   = 1    1 P   (2) N0(n) =  −  2 ψ  2 N0  −  2 + γ (N0  −  4) , N0   ≠   4.
3	Triangular (3, 3) {3}	${\begin{array}{l}\displaystyle {\frac {x}{(1-x)^{3}}}\end{array}}$	3	${\begin{array}{l}\displaystyle {1\,(n-1)+1}\end{array}}$ ${\begin{array}{l}\displaystyle {n\,}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(m+2)}\end{array}}$	${\begin{array}{l}\displaystyle {\frac {2m}{1+m}}\end{array}}$	${\begin{array}{l}\displaystyle {2(\psi (2)+\gamma )}\end{array}}$ ${\begin{array}{l}\displaystyle {2}\end{array}}$
4	Square (4, 4) {4}	${\begin{array}{l}\displaystyle {\frac {x\,(x+1)}{(1-x)^{3}}}\end{array}}$	4	${\begin{array}{l}\displaystyle {2\,(n-1)+1}\end{array}}$ ${\begin{array}{l}\displaystyle {2n-1}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(2m+1)}\end{array}}$	${\begin{array}{l}\displaystyle {H_{m}^{(2)}}\end{array}}$	${\begin{array}{l}\displaystyle {\zeta (2)={\frac {\pi ^{2}}{6}}}\end{array}}$ Base 10: A013661
5	Pentagonal (5, 5) {5}	${\begin{array}{l}\displaystyle {\frac {x\,(2x+1)}{(1-x)^{3}}}\end{array}}$	5	${\begin{array}{l}\displaystyle {3\,(n-1)+1}\end{array}}$ ${\begin{array}{l}\displaystyle {3n-2}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(3m-0)}\end{array}}$ ${\begin{array}{l}\displaystyle {{\frac {1}{2}}\,m^{2}(m+1)}\end{array}}$	${\begin{array}{l}\displaystyle {2\left(\psi \left(m+{\tfrac {2}{3}}\right)-\psi (m+1)-\psi \left({\tfrac {2}{3}}\right)-\gamma \right)}\end{array}}$	${\begin{array}{l}\displaystyle {-2\left(\psi \left({\tfrac {2}{3}}\right)+\gamma \right)}\end{array}}$ ${\begin{array}{l}\displaystyle {3\log(3)-{\frac {\pi \,{\sqrt {3}}}{3}}}\end{array}}$
6	Hexagonal (6, 6) {6}	${\begin{array}{l}\displaystyle {\frac {x\,(3x+1)}{(1-x)^{3}}}\end{array}}$	6	${\begin{array}{l}\displaystyle {4\,(n-1)+1}\end{array}}$ ${\begin{array}{l}\displaystyle {4n-3}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(4m-1)}\end{array}}$	${\begin{array}{l}\displaystyle {\left(\psi \left(m+{\tfrac {1}{2}}\right)-\psi (m+1)-\psi \left({\tfrac {1}{2}}\right)-\gamma \right)}\end{array}}$	${\begin{array}{l}\displaystyle {2\log(2)}\end{array}}$
7	Heptagonal (7, 7) {7}	${\begin{array}{l}\displaystyle {\frac {x\,(4x+1)}{(1-x)^{3}}}\end{array}}$	7	${\begin{array}{l}\displaystyle {5\,(n-1)+1}\end{array}}$ ${\begin{array}{l}\displaystyle {5n-4}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(5m-2)}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {2}{3}}\left(\psi \left(m+{\tfrac {2}{5}}\right)-\psi (m+1)-\psi \left({\tfrac {2}{5}}\right)-\gamma \right)}\end{array}}$	${\begin{array}{l}\displaystyle {-{\frac {2\left(\psi \left({\tfrac {2}{5}}\right)+\gamma \right)}{3}}}\end{array}}$
8	Octagonal (8, 8) {8}	${\begin{array}{l}\displaystyle {\frac {x\,(5x+1)}{(1-x)^{3}}}\end{array}}$	8	${\begin{array}{l}\displaystyle {6\,(n-1)+1}\end{array}}$ ${\begin{array}{l}\displaystyle {6n-5}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(6m-3)}\end{array}}$ ${\begin{array}{l}\displaystyle {{\frac {1}{2}}\,m\,(m+1)(2m-1)}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{2}}\left(\psi \left(m+{\tfrac {1}{3}}\right)-\psi (m+1)-\psi \left({\tfrac {1}{3}}\right)-\gamma \right)}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {3\log(3)}{4}}+{\frac {\pi {\sqrt {3}}}{12}}}\end{array}}$
9	9-gonal (9, 9) {9}	${\begin{array}{l}\displaystyle {\frac {x\,(6x+1)}{(1-x)^{3}}}\end{array}}$	9	${\begin{array}{l}\displaystyle {7\,(n-1)+1}\end{array}}$ ${\begin{array}{l}\displaystyle {7n-6}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(7m-4)}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {2}{5}}\left(\psi \left(m+{\tfrac {2}{7}}\right)-\psi (m+1)-\psi \left({\tfrac {2}{7}}\right)-\gamma \right)}\end{array}}$	${\begin{array}{l}\displaystyle {-{\frac {2\left(\psi \left({\tfrac {2}{7}}\right)+\gamma \right)}{5}}}\end{array}}$
10	10-gonal (10, 10) {10}	${\begin{array}{l}\displaystyle {\frac {x\,(7x+1)}{(1-x)^{3}}}\end{array}}$	10	${\begin{array}{l}\displaystyle {8\,(n-1)+1}\end{array}}$ ${\begin{array}{l}\displaystyle {8n-7}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(8m-5)}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{3}}\left(\psi \left(m+{\tfrac {1}{4}}\right)-\psi (m+1)-\psi \left({\tfrac {1}{4}}\right)-\gamma \right)}\end{array}}$	${\begin{array}{l}\displaystyle {\log(2)+{\frac {\pi }{6}}}\end{array}}$
11	11-gonal (11, 11) {11}	${\begin{array}{l}\displaystyle {\frac {x\,(8x+1)}{(1-x)^{3}}}\end{array}}$	11	${\begin{array}{l}\displaystyle {9\,(n-1)+1}\end{array}}$ ${\begin{array}{l}\displaystyle {9n-8}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(9m-6)}\end{array}}$ ${\begin{array}{l}\displaystyle {{\frac {1}{2}}\,m\,(m+1)(3m-2)}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {2}{7}}\left(\psi \left(m+{\tfrac {2}{9}}\right)-\psi (m+1)-\psi \left({\tfrac {2}{9}}\right)-\gamma \right)}\end{array}}$	${\begin{array}{l}\displaystyle {-{\frac {2\left(\psi \left({\tfrac {2}{9}}\right)+\gamma \right)}{7}}}\end{array}}$
12	12-gonal (12, 12) {12}	${\begin{array}{l}\displaystyle {\frac {x\,(9x+1)}{(1-x)^{3}}}\end{array}}$	12	${\begin{array}{l}\displaystyle {10\,(n-1)+1}\end{array}}$ ${\begin{array}{l}\displaystyle {10n-9}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(10m-7)}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{4}}\left(\psi \left(m+{\tfrac {1}{5}}\right)-\psi (m+1)-\psi \left({\tfrac {1}{5}}\right)-\gamma \right)}\end{array}}$	${\begin{array}{l}\displaystyle {\,}\end{array}}$
13	13-gonal (13, 13) {13}	${\begin{array}{l}\displaystyle {\frac {x\,(10x+1)}{(1-x)^{3}}}\end{array}}$	13	${\begin{array}{l}\displaystyle {11\,(n-1)+1}\end{array}}$ ${\begin{array}{l}\displaystyle {11n-10}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(11m-8)}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {2}{9}}\left(\psi \left(m+{\tfrac {2}{11}}\right)-\psi (m+1)-\psi \left({\tfrac {2}{11}}\right)-\gamma \right)}\end{array}}$	${\begin{array}{l}\displaystyle {-{\frac {2\left(\psi \left({\tfrac {2}{11}}\right)+\gamma \right)}{9}}}\end{array}}$
14	14-gonal (14, 14) {14}	${\begin{array}{l}\displaystyle {\frac {x\,(11x+1)}{(1-x)^{3}}}\end{array}}$	14	${\begin{array}{l}\displaystyle {12\,(n-1)+1}\end{array}}$ ${\begin{array}{l}\displaystyle {12n-11}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(12m-9)}\end{array}}$ ${\begin{array}{l}\displaystyle {{\frac {1}{2}}\,m\,(m+1)(4m-3)}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{5}}\left(\psi \left(m+{\tfrac {1}{6}}\right)-\psi (m+1)-\psi \left({\tfrac {1}{6}}\right)-\gamma \right)}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {2\log(2)}{5}}+{\frac {3\log(3)}{10}}+{\frac {\pi \,{\sqrt {3}}}{10}}}\end{array}}$
15	15-gonal (15, 15) {15}	${\begin{array}{l}\displaystyle {\frac {x\,(12x+1)}{(1-x)^{3}}}\end{array}}$	15	${\begin{array}{l}\displaystyle {13\,(n-1)+1}\end{array}}$ ${\begin{array}{l}\displaystyle {13n-12}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(13m-10)}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {2}{11}}\left(\psi \left(m+{\tfrac {2}{13}}\right)-\psi (m+1)-\psi \left({\tfrac {2}{13}}\right)-\gamma \right)}\end{array}}$	${\begin{array}{l}\displaystyle {-{\frac {2\left(\psi \left({\tfrac {2}{13}}\right)+\gamma \right)}{11}}}\end{array}}$
16	16-gonal (16, 16) {16}	${\begin{array}{l}\displaystyle {\frac {x\,(13x+1)}{(1-x)^{3}}}\end{array}}$	16	${\begin{array}{l}\displaystyle {14\,(n-1)+1}\end{array}}$ ${\begin{array}{l}\displaystyle {14n-13}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(14m-11)}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{6}}\left(\psi \left(m+{\tfrac {1}{7}}\right)-\psi (m+1)-\psi \left({\tfrac {1}{7}}\right)-\gamma \right)}\end{array}}$	${\begin{array}{l}\displaystyle {\,}\end{array}}$
17	17-gonal (17, 17) {17}	${\begin{array}{l}\displaystyle {\frac {x\,(14x+1)}{(1-x)^{3}}}\end{array}}$	17	${\begin{array}{l}\displaystyle {15\,(n-1)+1}\end{array}}$ ${\begin{array}{l}\displaystyle {15n-14}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(15m-12)}\end{array}}$ ${\begin{array}{l}\displaystyle {{\frac {1}{2}}\,m\,(m+1)(5m-4)}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {2}{13}}\left(\psi \left(m+{\tfrac {2}{15}}\right)-\psi (m+1)-\psi \left({\tfrac {2}{15}}\right)-\gamma \right)}\end{array}}$	${\begin{array}{l}\displaystyle {-{\frac {2\left(\psi \left({\tfrac {2}{15}}\right)+\gamma \right)}{13}}}\end{array}}$
18	18-gonal (18, 18) {18}	${\begin{array}{l}\displaystyle {\frac {x\,(15x+1)}{(1-x)^{3}}}\end{array}}$	18	${\begin{array}{l}\displaystyle {16\,(n-1)+1}\end{array}}$ ${\begin{array}{l}\displaystyle {16n-15}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(16m-13)}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{7}}\left(\psi \left(m+{\tfrac {1}{8}}\right)-\psi (m+1)-\psi \left({\tfrac {1}{8}}\right)-\gamma \right)}\end{array}}$	${\begin{array}{l}\displaystyle {\,}\end{array}}$
19	19-gonal (19, 19) {19}	${\begin{array}{l}\displaystyle {\frac {x\,(16x+1)}{(1-x)^{3}}}\end{array}}$	19	${\begin{array}{l}\displaystyle {17\,(n-1)+1}\end{array}}$ ${\begin{array}{l}\displaystyle {17n-16}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(17m-14)}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {2}{15}}\left(\psi \left(m+{\tfrac {2}{17}}\right)-\psi (m+1)-\psi \left({\tfrac {2}{17}}\right)-\gamma \right)}\end{array}}$	${\begin{array}{l}\displaystyle {-{\frac {2\left(\psi \left({\tfrac {2}{17}}\right)+\gamma \right)}{15}}}\end{array}}$
20	20-gonal (20, 20) {20}	${\begin{array}{l}\displaystyle {\frac {x\,(17x+1)}{(1-x)^{3}}}\end{array}}$	20	${\begin{array}{l}\displaystyle {18\,(n-1)+1}\end{array}}$ ${\begin{array}{l}\displaystyle {18n-17}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(18m-15)}\end{array}}$ ${\begin{array}{l}\displaystyle {{\frac {1}{2}}\,m\,(m+1)(6m-5)}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{8}}\left(\psi \left(m+{\tfrac {1}{9}}\right)-\psi (m+1)-\psi \left({\tfrac {1}{9}}\right)-\gamma \right)}\end{array}}$	${\begin{array}{l}\displaystyle {\,}\end{array}}$
21	21-gonal (21, 21) {21}	${\begin{array}{l}\displaystyle {\frac {x\,(18x+1)}{(1-x)^{3}}}\end{array}}$	21	${\begin{array}{l}\displaystyle {19\,(n-1)+1}\end{array}}$ ${\begin{array}{l}\displaystyle {19n-18}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(19m-16)}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {2}{17}}\left(\psi \left(m+{\tfrac {2}{19}}\right)-\psi (m+1)-\psi \left({\tfrac {2}{19}}\right)-\gamma \right)}\end{array}}$	${\begin{array}{l}\displaystyle {-{\frac {2\left(\psi \left({\tfrac {2}{19}}\right)+\gamma \right)}{17}}}\end{array}}$
22	22-gonal (22, 22) {22}	${\begin{array}{l}\displaystyle {\frac {x\,(19x+1)}{(1-x)^{3}}}\end{array}}$	22	${\begin{array}{l}\displaystyle {20\,(n-1)+1}\end{array}}$ ${\begin{array}{l}\displaystyle {20n-19}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(20m-17)}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{9}}\left(\psi \left(m+{\tfrac {1}{10}}\right)-\psi (m+1)-\psi \left({\tfrac {1}{10}}\right)-\gamma \right)}\end{array}}$	${\begin{array}{l}\displaystyle {\,}\end{array}}$
23	23-gonal (23, 23) {23}	${\begin{array}{l}\displaystyle {\frac {x\,(20x+1)}{(1-x)^{3}}}\end{array}}$	23	${\begin{array}{l}\displaystyle {21\,(n-1)+1}\end{array}}$ ${\begin{array}{l}\displaystyle {21n-20}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(21m-18)}\end{array}}$ ${\begin{array}{l}\displaystyle {{\frac {1}{2}}\,m\,(m+1)(7m-6)}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {2}{19}}\left(\psi \left(m+{\tfrac {2}{21}}\right)-\psi (m+1)-\psi \left({\tfrac {2}{21}}\right)-\gamma \right)}\end{array}}$	${\begin{array}{l}\displaystyle {-{\frac {2\left(\psi \left({\tfrac {2}{21}}\right)+\gamma \right)}{19}}}\end{array}}$
24	24-gonal (24, 24) {24}	${\begin{array}{l}\displaystyle {\frac {x\,(21x+1)}{(1-x)^{3}}}\end{array}}$	24	${\begin{array}{l}\displaystyle {22\,(n-1)+1}\end{array}}$ ${\begin{array}{l}\displaystyle {22n-21}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(22m-19)}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{10}}\left(\psi \left(m+{\tfrac {1}{11}}\right)-\psi (m+1)-\psi \left({\tfrac {1}{11}}\right)-\gamma \right)}\end{array}}$	${\begin{array}{l}\displaystyle {\,}\end{array}}$
25	25-gonal (25, 25) {25}	${\begin{array}{l}\displaystyle {\frac {x\,(22x+1)}{(1-x)^{3}}}\end{array}}$	25	${\begin{array}{l}\displaystyle {23\,(n-1)+1}\end{array}}$ ${\begin{array}{l}\displaystyle {23n-22}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(23m-20)}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {2}{21}}\left(\psi \left(m+{\tfrac {2}{23}}\right)-\psi (m+1)-\psi \left({\tfrac {2}{23}}\right)-\gamma \right)}\end{array}}$	${\begin{array}{l}\displaystyle {-{\frac {2\left(\psi \left({\tfrac {2}{23}}\right)+\gamma \right)}{21}}}\end{array}}$
26	26-gonal (26, 26) {26}	${\begin{array}{l}\displaystyle {\frac {x\,(23x+1)}{(1-x)^{3}}}\end{array}}$	26	${\begin{array}{l}\displaystyle {24\,(n-1)+1}\end{array}}$ ${\begin{array}{l}\displaystyle {24n-23}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(24m-21)}\end{array}}$ ${\begin{array}{l}\displaystyle {{\frac {1}{2}}\,m\,(m+1)(8m-7)}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{11}}\left(\psi \left(m+{\tfrac {1}{12}}\right)-\psi (m+1)-\psi \left({\tfrac {1}{12}}\right)-\gamma \right)}\end{array}}$	${\begin{array}{l}\displaystyle {\,}\end{array}}$
27	27-gonal (27, 27) {27}	${\begin{array}{l}\displaystyle {\frac {x\,(24x+1)}{(1-x)^{3}}}\end{array}}$	27	${\begin{array}{l}\displaystyle {25\,(n-1)+1}\end{array}}$ ${\begin{array}{l}\displaystyle {25n-24}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(25m-22)}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {2}{23}}\left(\psi \left(m+{\tfrac {2}{25}}\right)-\psi (m+1)-\psi \left({\tfrac {2}{25}}\right)-\gamma \right)}\end{array}}$	${\begin{array}{l}\displaystyle {-{\frac {2\left(\psi \left({\tfrac {2}{25}}\right)+\gamma \right)}{23}}}\end{array}}$
28	28-gonal (28, 28) {28}	${\begin{array}{l}\displaystyle {\frac {x\,(25x+1)}{(1-x)^{3}}}\end{array}}$	28	${\begin{array}{l}\displaystyle {26\,(n-1)+1}\end{array}}$ ${\begin{array}{l}\displaystyle {26n-25}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(26m-23)}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{12}}\left(\psi \left(m+{\tfrac {1}{13}}\right)-\psi (m+1)-\psi \left({\tfrac {1}{13}}\right)-\gamma \right)}\end{array}}$	${\begin{array}{l}\displaystyle {\,}\end{array}}$
29	29-gonal (29, 29) {29}	${\begin{array}{l}\displaystyle {\frac {x\,(26x+1)}{(1-x)^{3}}}\end{array}}$	29	${\begin{array}{l}\displaystyle {27\,(n-1)+1}\end{array}}$ ${\begin{array}{l}\displaystyle {27n-26}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(27m-24)\,}\end{array}}$ ${\begin{array}{l}\displaystyle {{\frac {1}{2}}\,m\,(m+1)(9m-8)}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {2}{25}}\left(\psi \left(m+{\tfrac {2}{27}}\right)-\psi (m+1)-\psi \left({\tfrac {2}{27}}\right)-\gamma \right)}\end{array}}$	${\begin{array}{l}\displaystyle {-{\frac {2\left(\psi \left({\tfrac {2}{27}}\right)+\gamma \right)}{25}}}\end{array}}$
30	30-gonal (30, 30) {30}	${\begin{array}{l}\displaystyle {\frac {x\,(27x+1)}{(1-x)^{3}}}\end{array}}$	30	${\begin{array}{l}\displaystyle {28\,(n-1)+1}\end{array}}$ ${\begin{array}{l}\displaystyle {28n-27}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(28m-25)}\end{array}}$	${\begin{array}{l}\displaystyle {{\frac {1}{13}}\left(\psi \left(m+{\tfrac {1}{14}}\right)-\psi (m+1)-\psi \left({\tfrac {1}{14}}\right)-\gamma \right)}\end{array}}$	${\begin{array}{l}\displaystyle {\,}\end{array}}$

Table of sequences

See also table of formulae and values above.

Polygonal numbers sequences

N0	P   (2) N0(n), n   ≥   0.	A-numbers
3	0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, ...	A000217
4	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, ...	A000290
5	0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, ...	A000326
6	0, 1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, 276, 325, 378, 435, 496, 561, 630, 703, 780, 861, 946, 1035, 1128, 1225, 1326, 1431, 1540, 1653, 1770, ...	A000384
7	0, 1, 7, 18, 34, 55, 81, 112, 148, 189, 235, 286, 342, 403, 469, 540, 616, 697, 783, 874, 970, 1071, 1177, 1288, 1404, 1525, 1651, 1782, 1918, 2059, ...	A000566
8	0, 1, 8, 21, 40, 65, 96, 133, 176, 225, 280, 341, 408, 481, 560, 645, 736, 833, 936, 1045, 1160, 1281, 1408, 1541, 1680, 1825, 1976, 2133, 2296, 2465, ...	A000567
9	0, 1, 9, 24, 46, 75, 111, 154, 204, 261, 325, 396, 474, 559, 651, 750, 856, 969, 1089, 1216, 1350, 1491, 1639, 1794, 1956, 2125, 2301, 2484, 2674, 2871, ...	A001106
10	0, 1, 10, 27, 52, 85, 126, 175, 232, 297, 370, 451, 540, 637, 742, 855, 976, 1105, 1242, 1387, 1540, 1701, 1870, 2047, 2232, 2425, 2626, 2835, 3052, ...	A001107
11	0, 1, 11, 30, 58, 95, 141, 196, 260, 333, 415, 506, 606, 715, 833, 960, 1096, 1241, 1395, 1558, 1730, 1911, 2101, 2300, 2508, 2725, 2951, 3186, 3430, ...	A051682
12	0, 1, 12, 33, 64, 105, 156, 217, 288, 369, 460, 561, 672, 793, 924, 1065, 1216, 1377, 1548, 1729, 1920, 2121, 2332, 2553, 2784, 3025, 3276, 3537, 3808, ...	A051624
13	0, 1, 13, 36, 70, 115, 171, 238, 316, 405, 505, 616, 738, 871, 1015, 1170, 1336, 1513, 1701, 1900, 2110, 2331, 2563, 2806, 3060, 3325, 3601, 3888, 4186, ...	A051865
14	0, 1, 14, 39, 76, 125, 186, 259, 344, 441, 550, 671, 804, 949, 1106, 1275, 1456, 1649, 1854, 2071, 2300, 2541, 2794, 3059, 3336, 3625, 3926, 4239, 4564, ...	A051866
15	0, 1, 15, 42, 82, 135, 201, 280, 372, 477, 595, 726, 870, 1027, 1197, 1380, 1576, 1785, 2007, 2242, 2490, 2751, 3025, 3312, 3612, 3925, 4251, 4590, ...	A051867
16	0, 1, 16, 45, 88, 145, 216, 301, 400, 513, 640, 781, 936, 1105, 1288, 1485, 1696, 1921, 2160, 2413, 2680, 2961, 3256, 3565, 3888, 4225, 4576, 4941, ...	A051868
17	0, 1, 17, 48, 94, 155, 231, 322, 428, 549, 685, 836, 1002, 1183, 1379, 1590, 1816, 2057, 2313, 2584, 2870, 3171, 3487, 3818, 4164, 4525, 4901, 5292, ...	A051869
18	0, 1, 18, 51, 100, 165, 246, 343, 456, 585, 730, 891, 1068, 1261, 1470, 1695, 1936, 2193, 2466, 2755, 3060, 3381, 3718, 4071, 4440, 4825, 5226, 5643, ...	A051870
19	0, 1, 19, 54, 106, 175, 261, 364, 484, 621, 775, 946, 1134, 1339, 1561, 1800, 2056, 2329, 2619, 2926, 3250, 3591, 3949, 4324, 4716, 5125, 5551, 5994, ...	A051871
20	0, 1, 20, 57, 112, 185, 276, 385, 512, 657, 820, 1001, 1200, 1417, 1652, 1905, 2176, 2465, 2772, 3097, 3440, 3801, 4180, 4577, 4992, 5425, 5876, 6345, ...	A051872
21	0, 1, 21, 60, 118, 195, 291, 406, 540, 693, 865, 1056, 1266, 1495, 1743, 2010, 2296, 2601, 2925, 3268, 3630, 4011, 4411, 4830, 5268, 5725, 6201, 6696, ...	A051873
22	0, 1, 22, 63, 124, 205, 306, 427, 568, 729, 910, 1111, 1332, 1573, 1834, 2115, 2416, 2737, 3078, 3439, 3820, 4221, 4642, 5083, 5544, 6025, 6526, 7047, ...	A051874
23	0, 1, 23, 66, 130, 215, 321, 448, 596, 765, 955, 1166, 1398, 1651, 1925, 2220, 2536, 2873, 3231, 3610, 4010, 4431, 4873, 5336, 5820, 6325, 6851, 7398, ...	A051875
24	0, 1, 24, 69, 136, 225, 336, 469, 624, 801, 1000, 1221, 1464, 1729, 2016, 2325, 2656, 3009, 3384, 3781, 4200, 4641, 5104, 5589, 6096, 6625, 7176, 7749, ...	A051876
25	0, 1, 25, 72, 142, 235, 351, 490, 652, 837, 1045, 1276, 1530, 1807, 2107, 2430, 2776, 3145, 3537, 3952, 4390, 4851, 5335, 5842, 6372, 6925, 7501, 8100, ...	A255184
26	0, 1, 26, 75, 148, 245, 366, 511, 680, 873, 1090, 1331, 1596, 1885, 2198, 2535, 2896, 3281, 3690, 4123, 4580, 5061, 5566, 6095, 6648, 7225, 7826, 8451, ...	A255185
27	0, 1, 27, 78, 154, 255, 381, 532, 708, 909, 1135, 1386, 1662, 1963, 2289, 2640, 3016, 3417, 3843, 4294, 4770, 5271, 5797, 6348, 6924, 7525, 8151, 8802, ...	A255186
28	0, 1, 28, 81, 160, 265, 396, 553, 736, 945, 1180, 1441, 1728, 2041, 2380, 2745, 3136, 3553, 3996, 4465, 4960, 5481, 6028, 6601, 7200, 7825, 8476, 9153, ...	A161935  (n  −  1), n   ≥   1
29	0, 1, 29, 84, 166, 275, 411, 574, 764, 981, 1225, 1496, 1794, 2119, 2471, 2850, 3256, 3689, 4149, 4636, 5150, 5691, 6259, 6854, 7476, 8125, 8801, 9504, ...	A255187
30	0, 1, 30, 87, 172, 285, 426, 595, 792, 1017, 1270, 1551, 1860, 2197, 2562, 2955, 3376, 3825, 4302, 4807, 5340, 5901, 6490, 7107, 7752, 8425, 9126, 9855, ...	A254474

Mysterious large overlap of terms of A176949 with terms in A140164

A176949:       4, 8, 14, 20, 26, 32, 38, 44, 50, 56, 62, 68, 74, 77, 80, 86,     98, 104, 110, 116, 119, 122, 128, 134, 140, 143, 146, 152, 158, 161, 164, 170,      182, 187, 188, 194, 200, 203, 206, 209, 212, 218, 221, 224, 230, 236, 242, 248, 254,      266, 272, 278, 284, 290, 296, 299, 302, ...

A140164: 1, 2, 4, 8, 14, 20, 26, 32, 38, 44, 50, 56, 62, 68, 74,     80, 86, 92, 98, 104, 110, 116,      122, 128, 134, 140,      146, 152, 158,      164, 170, 176, 182,      188, 194, 200,      206,      212, 218,      224, 230, 236, 242, 248, 254, 260, 266, 272, 278, 284, 290, 296,      302, ...

See also

• Centered polygonal numbers

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, 1994.