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Polygonal numbers

(Redirected from Pentagonal 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[2]
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
( jk  −  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
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

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

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

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

hence

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

${\displaystyle {\begin{array}{l}\displaystyle {r_{k}={\frac {(k-4)\pm {\sqrt {(k-4)^{2}+8n\,(k-2)}}}{2\,(k-2)}}.}\end{array}}}$
For example, the trigonal roots (i.e. triangular roots) of
 n
are
${\displaystyle {\begin{array}{l}\displaystyle {r_{3}={\frac {-1\pm {\sqrt {1+8n}}}{2}},}\end{array}}}$
the tetragonal roots (i.e. square roots) of
 n
are
${\displaystyle {\begin{array}{l}\displaystyle {r_{4}=\pm {\sqrt {n}},}\end{array}}}$
and the pentagonal roots of
 n
are
${\displaystyle {\begin{array}{l}\displaystyle {r_{5}={\frac {1\pm {\sqrt {1+24n}}}{6}}.}\end{array}}}$
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)
${\displaystyle {\begin{array}{l}\displaystyle {n=r\,\left[(k-2)\,r-(k-4)\right],\quad k=3,}\end{array}}}$

i.e.

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

yielding the oblong roots

${\displaystyle {\begin{array}{l}\displaystyle {r={\frac {-1\pm {\sqrt {1+4n}}}{2}}.}\end{array}}}$
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.[3]

For nondegenerate 2-dimensional regular convex polygons:

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

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

with initial conditions

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

Generating function

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

Order of basis

The order of basis of
 N0
-gonal numbers is:
${\displaystyle g_{\{P_{N_{0}}^{(2)}\}}=N_{0},\quad N_{0}\geq 3.\,}$
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}
 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[4]) 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[5]), 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[6] 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,[7] 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

${\displaystyle P_{N_{0}}^{(2)}(n)-P_{N_{0}}^{(2)}(n-1)=P_{N_{0}-1}^{(1)}(n)=(N_{0}-2)(n-1)+1,\,}$
where
 P (1) N0  −  1 (n)
is the
 n
th
 N0
-gonal gnomonic number.

Partial sums

${\displaystyle \sum _{n=1}^{m}P_{N_{0}}^{(2)}(n)=Y_{N_{0}+1}^{(3)}(m)={\frac {1}{6}}m(m+1)[(N_{0}-2)m-(N_{0}-5)]={\frac {1}{3}}P_{3}^{(2)}(m)[(N_{0}-2)m-(N_{0}-5)]={\frac {1}{3}}T_{m}[(N_{0}-2)m-(N_{0}-5)],\,}$
where
 Tm
is the
 m
th triangular number and
 Y  (3) N0  + 1 (m)
is the
 m
th
 N0
-gonal pyramidal number.[8]

Partial sums of reciprocals

For
 N0   ≠   4
,
{\displaystyle {\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,[9]
 γ
is the Euler–Mascheroni constant,[10] and
 ψ(x)
is the digamma function.[11] [12]
For
 N0 = 4
,
${\displaystyle \sum _{n=1}^{m}{\frac {1}{P_{4}^{(2)}(n)}}=H_{m}^{(2)}.\,}$

Sum of reciprocals

For
 N0   ≠   4
,
${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{P_{N_{0}}^{(2)}(n)}}=-{\frac {2\left(\psi \left({\frac {2}{N_{0}-2}}\right)+\gamma \right)}{(N_{0}-4)}}.\,}$
For
 N0 = 4
, the sum of reciprocals of the square numbers
${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{P_{4}^{(2)}(n)}}=\zeta (2)\,}$
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.[13]

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[14]

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
[5]
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 ψm +
 2 N0  −  2
−  ψ (m + 1)  −  ψ
 2 N0  −  2
−  γ
(N0  −  4)
,

 N0   ≠   4.
Sum of Reciprocals[15][16]

 ∞ ∑ n   = 1

 1 P   (2) N0(n)
=

−
2 ψ
 2 N0  −  2
+ γ
(N0  −  4)
,

 N0   ≠   4.
3 Triangular

(3, 3)

{3}

${\displaystyle {\begin{array}{l}\displaystyle {\frac {x}{(1-x)^{3}}}\end{array}}}$ 3 ${\displaystyle {\begin{array}{l}\displaystyle {1\,(n-1)+1}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {n\,}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(m+2)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {\frac {2m}{1+m}}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {2(\psi (2)+\gamma )}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {2}\end{array}}}$

4 Square

(4, 4)

{4}

${\displaystyle {\begin{array}{l}\displaystyle {\frac {x\,(x+1)}{(1-x)^{3}}}\end{array}}}$ 4 ${\displaystyle {\begin{array}{l}\displaystyle {2\,(n-1)+1}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {2n-1}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(2m+1)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {H_{m}^{(2)}}\end{array}}}$[17] ${\displaystyle {\begin{array}{l}\displaystyle {\zeta (2)={\frac {\pi ^{2}}{6}}}\end{array}}}$[18]

Base 10: A013661

5 Pentagonal

(5, 5)

{5}

${\displaystyle {\begin{array}{l}\displaystyle {\frac {x\,(2x+1)}{(1-x)^{3}}}\end{array}}}$ 5 ${\displaystyle {\begin{array}{l}\displaystyle {3\,(n-1)+1}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {3n-2}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(3m-0)}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{2}}\,m^{2}(m+1)}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {2\left(\psi \left(m+{\tfrac {2}{3}}\right)-\psi (m+1)-\psi \left({\tfrac {2}{3}}\right)-\gamma \right)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {-2\left(\psi \left({\tfrac {2}{3}}\right)+\gamma \right)}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {3\log(3)-{\frac {\pi \,{\sqrt {3}}}{3}}}\end{array}}}$

6 Hexagonal

(6, 6)

{6}

${\displaystyle {\begin{array}{l}\displaystyle {\frac {x\,(3x+1)}{(1-x)^{3}}}\end{array}}}$ 6 ${\displaystyle {\begin{array}{l}\displaystyle {4\,(n-1)+1}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {4n-3}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(4m-1)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {\left(\psi \left(m+{\tfrac {1}{2}}\right)-\psi (m+1)-\psi \left({\tfrac {1}{2}}\right)-\gamma \right)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {2\log(2)}\end{array}}}$
7 Heptagonal

(7, 7)

{7}

${\displaystyle {\begin{array}{l}\displaystyle {\frac {x\,(4x+1)}{(1-x)^{3}}}\end{array}}}$ 7 ${\displaystyle {\begin{array}{l}\displaystyle {5\,(n-1)+1}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {5n-4}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(5m-2)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {{\frac {2}{3}}\left(\psi \left(m+{\tfrac {2}{5}}\right)-\psi (m+1)-\psi \left({\tfrac {2}{5}}\right)-\gamma \right)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {-{\frac {2\left(\psi \left({\tfrac {2}{5}}\right)+\gamma \right)}{3}}}\end{array}}}$
8 Octagonal

(8, 8)

{8}

${\displaystyle {\begin{array}{l}\displaystyle {\frac {x\,(5x+1)}{(1-x)^{3}}}\end{array}}}$ 8 ${\displaystyle {\begin{array}{l}\displaystyle {6\,(n-1)+1}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {6n-5}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(6m-3)}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{2}}\,m\,(m+1)(2m-1)}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{2}}\left(\psi \left(m+{\tfrac {1}{3}}\right)-\psi (m+1)-\psi \left({\tfrac {1}{3}}\right)-\gamma \right)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {{\frac {3\log(3)}{4}}+{\frac {\pi {\sqrt {3}}}{12}}}\end{array}}}$
9 9-gonal

(9, 9)

{9}

${\displaystyle {\begin{array}{l}\displaystyle {\frac {x\,(6x+1)}{(1-x)^{3}}}\end{array}}}$ 9 ${\displaystyle {\begin{array}{l}\displaystyle {7\,(n-1)+1}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {7n-6}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(7m-4)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {{\frac {2}{5}}\left(\psi \left(m+{\tfrac {2}{7}}\right)-\psi (m+1)-\psi \left({\tfrac {2}{7}}\right)-\gamma \right)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {-{\frac {2\left(\psi \left({\tfrac {2}{7}}\right)+\gamma \right)}{5}}}\end{array}}}$
10 10-gonal

(10, 10)

{10}

${\displaystyle {\begin{array}{l}\displaystyle {\frac {x\,(7x+1)}{(1-x)^{3}}}\end{array}}}$ 10 ${\displaystyle {\begin{array}{l}\displaystyle {8\,(n-1)+1}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {8n-7}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(8m-5)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{3}}\left(\psi \left(m+{\tfrac {1}{4}}\right)-\psi (m+1)-\psi \left({\tfrac {1}{4}}\right)-\gamma \right)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {\log(2)+{\frac {\pi }{6}}}\end{array}}}$
11 11-gonal

(11, 11)

{11}

${\displaystyle {\begin{array}{l}\displaystyle {\frac {x\,(8x+1)}{(1-x)^{3}}}\end{array}}}$ 11 ${\displaystyle {\begin{array}{l}\displaystyle {9\,(n-1)+1}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {9n-8}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(9m-6)}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{2}}\,m\,(m+1)(3m-2)}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {2}{7}}\left(\psi \left(m+{\tfrac {2}{9}}\right)-\psi (m+1)-\psi \left({\tfrac {2}{9}}\right)-\gamma \right)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {-{\frac {2\left(\psi \left({\tfrac {2}{9}}\right)+\gamma \right)}{7}}}\end{array}}}$
12 12-gonal

(12, 12)

{12}

${\displaystyle {\begin{array}{l}\displaystyle {\frac {x\,(9x+1)}{(1-x)^{3}}}\end{array}}}$ 12 ${\displaystyle {\begin{array}{l}\displaystyle {10\,(n-1)+1}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {10n-9}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(10m-7)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{4}}\left(\psi \left(m+{\tfrac {1}{5}}\right)-\psi (m+1)-\psi \left({\tfrac {1}{5}}\right)-\gamma \right)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {\,}\end{array}}}$
13 13-gonal

(13, 13)

{13}

${\displaystyle {\begin{array}{l}\displaystyle {\frac {x\,(10x+1)}{(1-x)^{3}}}\end{array}}}$ 13 ${\displaystyle {\begin{array}{l}\displaystyle {11\,(n-1)+1}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {11n-10}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(11m-8)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {{\frac {2}{9}}\left(\psi \left(m+{\tfrac {2}{11}}\right)-\psi (m+1)-\psi \left({\tfrac {2}{11}}\right)-\gamma \right)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {-{\frac {2\left(\psi \left({\tfrac {2}{11}}\right)+\gamma \right)}{9}}}\end{array}}}$
14 14-gonal

(14, 14)

{14}

${\displaystyle {\begin{array}{l}\displaystyle {\frac {x\,(11x+1)}{(1-x)^{3}}}\end{array}}}$ 14 ${\displaystyle {\begin{array}{l}\displaystyle {12\,(n-1)+1}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {12n-11}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(12m-9)}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{2}}\,m\,(m+1)(4m-3)}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{5}}\left(\psi \left(m+{\tfrac {1}{6}}\right)-\psi (m+1)-\psi \left({\tfrac {1}{6}}\right)-\gamma \right)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {{\frac {2\log(2)}{5}}+{\frac {3\log(3)}{10}}+{\frac {\pi \,{\sqrt {3}}}{10}}}\end{array}}}$
15 15-gonal

(15, 15)

{15}

${\displaystyle {\begin{array}{l}\displaystyle {\frac {x\,(12x+1)}{(1-x)^{3}}}\end{array}}}$ 15 ${\displaystyle {\begin{array}{l}\displaystyle {13\,(n-1)+1}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {13n-12}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(13m-10)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {{\frac {2}{11}}\left(\psi \left(m+{\tfrac {2}{13}}\right)-\psi (m+1)-\psi \left({\tfrac {2}{13}}\right)-\gamma \right)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {-{\frac {2\left(\psi \left({\tfrac {2}{13}}\right)+\gamma \right)}{11}}}\end{array}}}$
16 16-gonal

(16, 16)

{16}

${\displaystyle {\begin{array}{l}\displaystyle {\frac {x\,(13x+1)}{(1-x)^{3}}}\end{array}}}$ 16 ${\displaystyle {\begin{array}{l}\displaystyle {14\,(n-1)+1}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {14n-13}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(14m-11)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{6}}\left(\psi \left(m+{\tfrac {1}{7}}\right)-\psi (m+1)-\psi \left({\tfrac {1}{7}}\right)-\gamma \right)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {\,}\end{array}}}$
17 17-gonal

(17, 17)

{17}

${\displaystyle {\begin{array}{l}\displaystyle {\frac {x\,(14x+1)}{(1-x)^{3}}}\end{array}}}$ 17 ${\displaystyle {\begin{array}{l}\displaystyle {15\,(n-1)+1}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {15n-14}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(15m-12)}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{2}}\,m\,(m+1)(5m-4)}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {2}{13}}\left(\psi \left(m+{\tfrac {2}{15}}\right)-\psi (m+1)-\psi \left({\tfrac {2}{15}}\right)-\gamma \right)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {-{\frac {2\left(\psi \left({\tfrac {2}{15}}\right)+\gamma \right)}{13}}}\end{array}}}$
18 18-gonal

(18, 18)

{18}

${\displaystyle {\begin{array}{l}\displaystyle {\frac {x\,(15x+1)}{(1-x)^{3}}}\end{array}}}$ 18 ${\displaystyle {\begin{array}{l}\displaystyle {16\,(n-1)+1}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {16n-15}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(16m-13)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{7}}\left(\psi \left(m+{\tfrac {1}{8}}\right)-\psi (m+1)-\psi \left({\tfrac {1}{8}}\right)-\gamma \right)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {\,}\end{array}}}$
19 19-gonal

(19, 19)

{19}

${\displaystyle {\begin{array}{l}\displaystyle {\frac {x\,(16x+1)}{(1-x)^{3}}}\end{array}}}$ 19 ${\displaystyle {\begin{array}{l}\displaystyle {17\,(n-1)+1}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {17n-16}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(17m-14)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {{\frac {2}{15}}\left(\psi \left(m+{\tfrac {2}{17}}\right)-\psi (m+1)-\psi \left({\tfrac {2}{17}}\right)-\gamma \right)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {-{\frac {2\left(\psi \left({\tfrac {2}{17}}\right)+\gamma \right)}{15}}}\end{array}}}$
20 20-gonal

(20, 20)

{20}

${\displaystyle {\begin{array}{l}\displaystyle {\frac {x\,(17x+1)}{(1-x)^{3}}}\end{array}}}$ 20 ${\displaystyle {\begin{array}{l}\displaystyle {18\,(n-1)+1}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {18n-17}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(18m-15)}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{2}}\,m\,(m+1)(6m-5)}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{8}}\left(\psi \left(m+{\tfrac {1}{9}}\right)-\psi (m+1)-\psi \left({\tfrac {1}{9}}\right)-\gamma \right)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {\,}\end{array}}}$
21 21-gonal

(21, 21)

{21}

${\displaystyle {\begin{array}{l}\displaystyle {\frac {x\,(18x+1)}{(1-x)^{3}}}\end{array}}}$ 21 ${\displaystyle {\begin{array}{l}\displaystyle {19\,(n-1)+1}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {19n-18}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(19m-16)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {{\frac {2}{17}}\left(\psi \left(m+{\tfrac {2}{19}}\right)-\psi (m+1)-\psi \left({\tfrac {2}{19}}\right)-\gamma \right)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {-{\frac {2\left(\psi \left({\tfrac {2}{19}}\right)+\gamma \right)}{17}}}\end{array}}}$
22 22-gonal

(22, 22)

{22}

${\displaystyle {\begin{array}{l}\displaystyle {\frac {x\,(19x+1)}{(1-x)^{3}}}\end{array}}}$ 22 ${\displaystyle {\begin{array}{l}\displaystyle {20\,(n-1)+1}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {20n-19}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(20m-17)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{9}}\left(\psi \left(m+{\tfrac {1}{10}}\right)-\psi (m+1)-\psi \left({\tfrac {1}{10}}\right)-\gamma \right)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {\,}\end{array}}}$
23 23-gonal

(23, 23)

{23}

${\displaystyle {\begin{array}{l}\displaystyle {\frac {x\,(20x+1)}{(1-x)^{3}}}\end{array}}}$ 23 ${\displaystyle {\begin{array}{l}\displaystyle {21\,(n-1)+1}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {21n-20}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(21m-18)}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{2}}\,m\,(m+1)(7m-6)}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {2}{19}}\left(\psi \left(m+{\tfrac {2}{21}}\right)-\psi (m+1)-\psi \left({\tfrac {2}{21}}\right)-\gamma \right)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {-{\frac {2\left(\psi \left({\tfrac {2}{21}}\right)+\gamma \right)}{19}}}\end{array}}}$
24 24-gonal

(24, 24)

{24}

${\displaystyle {\begin{array}{l}\displaystyle {\frac {x\,(21x+1)}{(1-x)^{3}}}\end{array}}}$ 24 ${\displaystyle {\begin{array}{l}\displaystyle {22\,(n-1)+1}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {22n-21}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(22m-19)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{10}}\left(\psi \left(m+{\tfrac {1}{11}}\right)-\psi (m+1)-\psi \left({\tfrac {1}{11}}\right)-\gamma \right)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {\,}\end{array}}}$
25 25-gonal

(25, 25)

{25}

${\displaystyle {\begin{array}{l}\displaystyle {\frac {x\,(22x+1)}{(1-x)^{3}}}\end{array}}}$ 25 ${\displaystyle {\begin{array}{l}\displaystyle {23\,(n-1)+1}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {23n-22}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(23m-20)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {{\frac {2}{21}}\left(\psi \left(m+{\tfrac {2}{23}}\right)-\psi (m+1)-\psi \left({\tfrac {2}{23}}\right)-\gamma \right)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {-{\frac {2\left(\psi \left({\tfrac {2}{23}}\right)+\gamma \right)}{21}}}\end{array}}}$
26 26-gonal

(26, 26)

{26}

${\displaystyle {\begin{array}{l}\displaystyle {\frac {x\,(23x+1)}{(1-x)^{3}}}\end{array}}}$ 26 ${\displaystyle {\begin{array}{l}\displaystyle {24\,(n-1)+1}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {24n-23}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(24m-21)}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{2}}\,m\,(m+1)(8m-7)}\end{array}}}$
${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{11}}\left(\psi \left(m+{\tfrac {1}{12}}\right)-\psi (m+1)-\psi \left({\tfrac {1}{12}}\right)-\gamma \right)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {\,}\end{array}}}$
27 27-gonal

(27, 27)

{27}

${\displaystyle {\begin{array}{l}\displaystyle {\frac {x\,(24x+1)}{(1-x)^{3}}}\end{array}}}$ 27 ${\displaystyle {\begin{array}{l}\displaystyle {25\,(n-1)+1}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {25n-24}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(25m-22)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {{\frac {2}{23}}\left(\psi \left(m+{\tfrac {2}{25}}\right)-\psi (m+1)-\psi \left({\tfrac {2}{25}}\right)-\gamma \right)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {-{\frac {2\left(\psi \left({\tfrac {2}{25}}\right)+\gamma \right)}{23}}}\end{array}}}$
28 28-gonal

(28, 28)

{28}

${\displaystyle {\begin{array}{l}\displaystyle {\frac {x\,(25x+1)}{(1-x)^{3}}}\end{array}}}$ 28 ${\displaystyle {\begin{array}{l}\displaystyle {26\,(n-1)+1}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {26n-25}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(26m-23)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{12}}\left(\psi \left(m+{\tfrac {1}{13}}\right)-\psi (m+1)-\psi \left({\tfrac {1}{13}}\right)-\gamma \right)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {\,}\end{array}}}$
29 29-gonal

(29, 29)

{29}

${\displaystyle {\begin{array}{l}\displaystyle {\frac {x\,(26x+1)}{(1-x)^{3}}}\end{array}}}$ 29 ${\displaystyle {\begin{array}{l}\displaystyle {27\,(n-1)+1}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {27n-26}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(27m-24)\,}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{2}}\,m\,(m+1)(9m-8)}\end{array}}}$
${\displaystyle {\begin{array}{l}\displaystyle {{\frac {2}{25}}\left(\psi \left(m+{\tfrac {2}{27}}\right)-\psi (m+1)-\psi \left({\tfrac {2}{27}}\right)-\gamma \right)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {-{\frac {2\left(\psi \left({\tfrac {2}{27}}\right)+\gamma \right)}{25}}}\end{array}}}$
30 30-gonal

(30, 30)

{30}

${\displaystyle {\begin{array}{l}\displaystyle {\frac {x\,(27x+1)}{(1-x)^{3}}}\end{array}}}$ 30 ${\displaystyle {\begin{array}{l}\displaystyle {28\,(n-1)+1}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {28n-27}\end{array}}}$

${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{6}}\,m\,(m+1)(28m-25)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {{\frac {1}{13}}\left(\psi \left(m+{\tfrac {1}{14}}\right)-\psi (m+1)-\psi \left({\tfrac {1}{14}}\right)-\gamma \right)}\end{array}}}$ ${\displaystyle {\begin{array}{l}\displaystyle {\,}\end{array}}}$

Table of sequences

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, ...


Notes

1. Author of the plots: Stefan Friedrich Birkner, License: Creative Commons Attribution-ShareAlike 3.0 Unported.
2. Where
 P   (d ) N0 (n)
is the
 d
-dimensional regular convex polytope number with
 N0
0-dimensional facets, i.e. vertices
 V
.
3. Weisstein, Eric W., Polyhedral Formula, from MathWorld—A Wolfram Web Resource.
4. Weisstein, Eric W., Lagrange's Four-Square Theorem, from MathWorld—A Wolfram Web Resource.
5. Weisstein, Eric W., Fermat's Polygonal Number Theorem, from MathWorld—A Wolfram Web Resource.
6. Weisstein, Eric W., Waring's Problem, from MathWorld—A Wolfram Web Resource.
7. Weisstein, Eric W., Fifteen Theorem, from MathWorld—A Wolfram Web Resource.
8. Where
 Y  (d ) [(k  + 2) + (d  − 2)]  (n) = Y  (d ) k  + d  (n), k   ≥   1, n   ≥   0,
is the
 d
-dimensional,
 d   ≥   0
,
 (k  +  2)
-gonal base (hyper)pyramidal number where, for
 d   ≥   2
,
 N0 = [(k  +  2)  +  (d  −  2)]
is the number of vertices (including the
 d  −  2
apex vertices) of the polygonal base (hyper)pyramid.
9. Sondow, Jonathan and Weisstein, Eric W., Harmonic Number, from MathWorld—A Wolfram Web Resource, created by Eric W. Weisstein.
10. Weisstein, Eric W., Euler-Mascheroni Constant, from MathWorld—A Wolfram Web Resource.
11. Weisstein, Eric W., Digamma Function, from MathWorld—A Wolfram Web Resource.
12. Weisstein, Eric W., Polygamma Function, from MathWorld—A Wolfram Web Resource.
13. Weisstein, Eric W., Relatively Prime, from MathWorld—A Wolfram Web Resource.
14. Weisstein, Eric W., Schläfli Symbol, from MathWorld—A Wolfram Web Resource.
15. Downey, Lawrence M., Ong, Boon W., and Sellers, James A., Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, 2008.
16. Psychedelic Geometry, Inverse Polygonal Numbers Series.
17. 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.
18. Sondow, Jonathan and Weisstein, Eric W., Riemann Zeta Function, from MathWorld—A Wolfram Web Resource, created by Eric W. Weisstein.