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Polygonal numbers

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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+1 dots for each edge (including both end vertices) of the nth layer, n ≥ 1, with all layers sharing a common vertex (which corresponds to n = 0) and the 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

Contents

Formulae

The nth N0-gonal number is given by the formulae:[2]

P^{(2)}_{N_0}(n) \equiv \sum_{i=0}^{n} P^{(1)}_{N_0-1}(i)\,
P^{(2)}_{N_0}(n) = n + (N_0-2) P^{(2)}_{3}(n-1) = n + (N_0-2) T_{n-1} = n + (N_0-2) \binom{n}{2} = n + (N_0-2) {(n-1)n\over2} = {\frac{n}{2}}[(N_0-2)n - (N_0-4)],\,

where \scriptstyle P^{(1)}_{N_0}(n)\, is the nth N0-gonal gnomonic number, and where N0 is the number of 0-dimensional elements (which are vertices V) of the polygons and \scriptstyle T_n\, is the nth triangular number.

Schläfli-Poincaré (convex) polytope formula

Schläfli-Poincaré generalization of the Descartes-Euler (convex) polyhedral formula.[3]

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^{(2)}_{N_0}(n) = 3P^{(2)}_{N_0}(n-1) - 3P^{(2)}_{N_0}(n-2) + P^{(2)}_{N_0}(n-3),\ n>2,\,

with initial conditions

P^{(2)}_{N_0}(0) = 0,\ P^{(2)}_{N_0}(1) = 1,\ P^{(2)}_{N_0}(2) = N_0\,

Generating function

G_{\{P^{(2)}_{N_0}(n)\}}(x) = {{x((N_0 - 3)x+1)}\over{(1-x)^3}}\,

Order of basis

The order of basis of N0-gonal numbers is:

g_{\{P^{(2)}_{N_0}\}} = N_0,\ N_0 \ge 3.\,

The order of basis g for numbers of the form \scriptstyle kn+1,\ k > 0\, is k, since to represent the numbers in the congruence classes \scriptstyle \{0, 1, ..., k-1\}\, by adding numbers congruent to \scriptstyle 1 \mod k\, we need as many terms as the class number, for each congruence classes, e.g. for \scriptstyle k = 5\,:

numbers of form \scriptstyle 5n+1\, are expressible as 1 term of the form \scriptstyle 5n+1\,;
numbers of form \scriptstyle 5n+2\, are expressible as the sum of 2 terms of the form \scriptstyle 5n+1\,;
numbers of form \scriptstyle 5n+3\, are expressible as the sum of 3 terms of the form \scriptstyle 5n+1\,;
numbers of form \scriptstyle 5n+4\, are expressible as the sum of 4 terms of the form \scriptstyle 5n+1\,;
numbers of form \scriptstyle 5n+0\, are expressible as the sum of 5 terms of the form \scriptstyle 5n+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-polygonal 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-gon numbers (known as the polygonal number theorem,[4]) 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 \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 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 \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[5] 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 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,[6] 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^{(2)}_{N_0}(n) - P^{(2)}_{N_0}(n-1) = P^{(1)}_{N_0-1}(n) = (N_0 - 2) (n-1) + 1,\,

where \scriptstyle P^{(1)}(N_0-1, n)\, is the nth N0-gonal gnomonic number.

Partial sums

\sum_{n=1}^m P^{(2)}_{N_0}(n) = Y^{(3)}_{N_0+1}(m) = \frac{1}{6} m (m+1) [(N_0 - 2) m - (N_0 - 5)] = \frac{1}{3} P^{(2)}_{3}(m) [(N_0 - 2) m - (N_0 - 5)] = \frac{1}{3} T_m [(N_0 - 2) m - (N_0 - 5)],\,

where \scriptstyle T_m\, is the mth triangular number and \scriptstyle Y^{(3)}_{N_0+1}(m)\, is the mth N0-gonal pyramidal number. [7]

Partial sums of reciprocals

For \scriptstyle N_0 \neq 4\,:

\sum_{n=1}^m \frac{1}{P^{(2)}_{N_0}(n)} = - \frac{2 \big(H_m - \psi(m + \frac{2}{N_0-2})+\psi(\frac{2}{N_0-2})\big)}{(N_0-4)} = - \frac{2 \big(\psi(m+1) + \gamma -\psi(m + \frac{2}{N_0-2}) + \psi(\frac{2}{N_0-2})\big)}{(N_0-4)},\,
= \frac{2 \big(\psi(m + \frac{2}{N_0-2}) -\psi(m+1) - \psi(\frac{2}{N_0-2}) - \gamma\big)}{(N_0-4)}\,

where \scriptstyle H_m\, is the mth harmonic number,[8] \scriptstyle \gamma\, is the Euler-Mascheroni constant,[9] and \scriptstyle \psi(x)\, is the digamma function.[10] [11]

For \scriptstyle N_0 = 4\,:

\sum_{n=1}^m \frac{1}{P^{(2)}_{4}(n)} = H_{m}^{(2)}\,

Sum of reciprocals

For \scriptstyle N_0 \neq 4\,:

\sum_{n=1}^\infty \frac{1}{P^{(2)}_{N_0}(n)} = - \frac{2\big(\psi\big(\frac{2}{N_0-2}\big)+\gamma\big)}{(N_0-4)}\,

For \scriptstyle N_0 = 4\,, the sum of reciprocals of the square numbers:

{\sum_{n=1}^\infty {1\over{P^{(2)}_{4}(n)}}} = \zeta(2)

can be interpreted as \scriptstyle \frac{1}{p}, where \scriptstyle p = {\frac{1}{\zeta(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 xy is squarefree.[12]

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
N_0 \, Name Formulae

P^{(2)}_{N_0}(n) = \,


{\frac{n}{2}[(N_0-2)n - (N_0-4)]} \,

n = 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 + 2P^{(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 A??????
26 26-gonal n(12n-11) \, 0 1 26 75 148 245 366 511 680 873 1090 1331 1596 A??????
27 27-gonal n(25n-23)/2 \, 0 1 27 78 154 255 381 532 708 909 1135 1386 1662 A??????
28 28-gonal n(13n-12) \, 0 1 28 81 160 265 396 553 736 945 1180 1441 1728 A??????
29 29-gonal n(27n-25)/2 \, 0 1 29 84 166 275 411 574 764 981 1225 1496 1794 A??????
30 30-gonal n(14n-13) \, 0 1 30 87 172 285 426 595 792 1017 1270 1551 1860 A??????


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) are named in bold.

Polygonal numbers related formulae and values
N0 Name

(N0, N1)

Schläfli symbol[13]

Generating
function

G_{\{P^{(2)}_{N_0}(n)\}}(x)


\scriptstyle = \frac{x \, ((N_0 - 3) \, x + 1)}{(1-x)^3}

Order
of basis

\scriptstyle g_{ \{ P^{(2)}_{N_0} \} }

= N0,


N_0 \ge 3[4]

Differences

P^{(2)}_{N_0}(n) -

P^{(2)}_{N_0}(n-1) =


P^{(1)}_{N_0 - 1}(n) =


\scriptstyle (N_0 - 2) (n-1) + 1

Gnomonic numbers

Partial sums


\sum_{n=1}^{m} P^{(2)}_{N_0}(n) =


Y^{(3)}_{N_0 + 1}(m) =


\scriptstyle \frac{T_m}{3} \, [(N_0 - 2) m - (N_0 - 5)]

Partial sums of reciprocals

\sum_{n=1}^{m} \frac{1}{ P^{(2)}_{N_0}(n)} =


\scriptstyle \frac{2 \left( \psi\left( m + \frac{2}{N_0 - 2} \right) - \psi(m+1) - \psi\left( \frac{2}{N_0 - 2} \right) - \gamma \right) }{(N_0-4)},


N_0 \neq 4.

Sum of Reciprocals[14][15]

\sum_{n=1}^{\infty} \frac{1}{ P^{(2)}_{N_0}(n) } =


\scriptstyle  - \frac{2 \left( \psi\left( \frac{2}{N_0 - 2} \right) + \gamma \right) }{(N_0 - 4)},


N_0 \neq 4.

3 Triangular

(3, 3)

{3}

\scriptstyle \frac{x}{(1-x)^3} 3 \scriptstyle 1(n-1) + 1

n\,

\scriptstyle \frac{1}{6} \, m \, (m+1)(m+2) \frac{2 m}{(1 + m)} 2(ψ(2) + γ)


2

4 Square

(4, 4)

{4}

\scriptstyle \frac{x \, (x+1)}{(1-x)^3} 4 \scriptstyle 2 \, (n-1) + 1

\scriptstyle 2n - 1

\scriptstyle \frac{1}{6} \, m \, (m+1)(2m+1) H_{m}^{(2)} [16] \zeta(2) = \frac{\pi^2}{6}[17]

Base 10: A013661

5 Pentagonal

(5, 5)

{5}

\scriptstyle \frac{x \, (2x+1)}{(1-x)^3} 5 \scriptstyle 3 \, (n-1) + 1

\scriptstyle 3n - 2

\scriptstyle \frac{1}{6} \, m \, (m+1)(3m-0)


\scriptstyle \frac{1}{2} \, m^2 (m+1)

\scriptstyle 2 \left(\psi\left( m + \tfrac{2}{3} \right) - \psi(m+1) - \psi\left(\tfrac{2}{3}\right) - \gamma \right)  - 2 \left(\psi\left(\tfrac{2}{3}\right) + \gamma \right)

3 \log(3) - \frac{\pi \, \sqrt{3}}{3}

6 Hexagonal

(6, 6)

{6}

\scriptstyle \frac{x \, (3x+1)}{(1-x)^3} 6 \scriptstyle 4 \, (n-1) + 1

\scriptstyle 4n - 3

\scriptstyle \frac{1}{6} \, m \, (m+1)(4m-1) \scriptstyle \left( \psi\left( m + \tfrac{1}{2} \right) - \psi(m+1) - \psi\left(\tfrac{1}{2}\right) - \gamma \right) 2log(2)
7 Heptagonal

(7, 7)

{7}

\scriptstyle \frac{x \, (4x+1)}{(1-x)^3} 7 \scriptstyle 5 \, (n-1) + 1

\scriptstyle 5n - 4

\scriptstyle \frac{1}{6} \, m \, (m+1) (5m-2) \scriptstyle \frac{2}{3} \left( \psi\left( m + \tfrac{2}{5} \right) - \psi(m+1) - \psi\left(\tfrac{2}{5}\right) - \gamma \right)  - \frac{2 \left( \psi\left( \tfrac{2}{5} \right) + \gamma \right) }{3}
8 Octagonal

(8, 8)

{8}

\scriptstyle \frac{x \, (5x+1)}{(1-x)^3} 8 \scriptstyle 6 \, (n-1) + 1

\scriptstyle 6n - 5

\scriptstyle \frac{1}{6} \, m \, (m+1) (6m-3)


\scriptstyle \frac{1}{2} \, m \, (m+1) (2m-1)

\scriptstyle \frac{1}{2} \left( \psi\left( m + \tfrac{1}{3} \right) - \psi(m+1) - \psi\left(\tfrac{1}{3}\right) - \gamma \right) \frac{3 \log(3)}{4} + \frac{\pi \sqrt{3}}{12}
9 9-gonal

(9, 9)

{9}

\scriptstyle \frac{x \, (6x+1)}{(1-x)^3} 9 \scriptstyle 7 \, (n-1) + 1

\scriptstyle 7n - 6

\scriptstyle \frac{1}{6} \, m \, (m+1) (7m-4) \scriptstyle \frac{2}{5} \left( \psi\left( m + \tfrac{2}{7} \right) - \psi(m+1)- \psi\left(\tfrac{2}{7}\right) - \gamma \right)  - \frac{ 2 \left( \psi\left( \tfrac{2}{7} \right) + \gamma \right) }{5}
10 10-gonal

(10, 10)

{10}

\scriptstyle \frac{x \, (7x+1)}{(1-x)^3} 10 \scriptstyle 8 \, (n-1) + 1

\scriptstyle 8n - 7

\scriptstyle \frac{1}{6} \, m \, (m+1) (8m-5) \scriptstyle \frac{1}{3} \left( \psi\left( m + \tfrac{1}{4} \right) - \psi(m+1)- \psi\left(\tfrac{1}{4}\right) - \gamma \right) \log(2) + \frac{\pi}{6}
11 11-gonal

(11, 11)

{11}

\scriptstyle \frac{x \, (8x+1)}{(1-x)^3} 11 \scriptstyle 9 \, (n-1) + 1

\scriptstyle 9n - 8

\scriptstyle \frac{1}{6} \, m \, (m+1) (9m-6)


\scriptstyle \frac{1}{2} \, m \, (m+1) (3m-2)

\scriptstyle \frac{2}{7} \left( \psi\left( m + \tfrac{2}{9} \right) - \psi(m+1) - \psi\left(\tfrac{2}{9}\right) - \gamma \right)  - \frac{ 2 \left( \psi\left( \tfrac{2}{9} \right) + \gamma \right) }{7}
12 12-gonal

(12, 12)

{12}

\scriptstyle \frac{x \, (9x+1)}{(1-x)^3} 12 \scriptstyle 10 \, (n-1) + 1

\scriptstyle 10n - 9

\scriptstyle \frac{1}{6} \, m \, (m+1) (10m-7) \scriptstyle \frac{1}{4} \left(\psi\left( m + \tfrac{1}{5} \right) - \psi(m+1) - \psi\left(\tfrac{1}{5}\right) - \gamma \right) \,
13 13-gonal

(13, 13)

{13}

\scriptstyle \frac{x \, (10x+1)}{(1-x)^3} 13 \scriptstyle 11 \, (n-1) + 1

\scriptstyle 11n - 10

\scriptstyle \frac{1}{6} \, m \, (m+1) (11m-8) \scriptstyle \frac{2}{9} \left( \psi\left( m + \tfrac{2}{11} \right) - \psi(m+1)- \psi\left(\tfrac{2}{11}\right) - \gamma \right)  - \frac{ 2 \left( \psi\left( \tfrac{2}{11} \right) + \gamma \right) }{9}
14 14-gonal

(14, 14)

{14}

\scriptstyle \frac{x \, (11x+1)}{(1-x)^3} 14 \scriptstyle 12 \, (n-1) + 1

\scriptstyle 12n - 11

\scriptstyle \frac{1}{6} \, m \, (m+1) (12m-9)


\scriptstyle \frac{1}{2} \, m \, (m+1) (4m-3)

\scriptstyle \frac{1}{5} \left( \psi\left( m + \tfrac{1}{6} \right) - \psi(m+1) - \psi\left(\tfrac{1}{6}\right) - \gamma \right) \frac{2 \log(2)}{5} + \frac{3 \log(3)}{10} + \frac{\pi \, \sqrt{3}}{10}
15 15-gonal

(15, 15)

{15}

\scriptstyle \frac{x \, (12x+1)}{(1-x)^3} 15 \scriptstyle 13 \, (n-1) + 1

\scriptstyle 13n - 12

\scriptstyle \frac{1}{6} \, m \, (m+1) (13m-10) \scriptstyle \frac{2}{11} \left( \psi\left( m + \tfrac{2}{13} \right) - \psi(m+1) - \psi\left(\tfrac{2}{13}\right) - \gamma \right)  - \frac{ 2 \left( \psi\left( \tfrac{2}{13} \right) + \gamma \right) }{11}
16 16-gonal

(16, 16)

{16}

\scriptstyle \frac{x \, (13x+1)}{(1-x)^3} 16 \scriptstyle 14 \, (n-1) + 1

\scriptstyle 14n - 13

\scriptstyle \frac{1}{6} \, m \, (m+1) (14m-11) \scriptstyle \frac{1}{6} \left( \psi\left( m + \tfrac{1}{7} \right) - \psi(m+1) - \psi\left(\tfrac{1}{7}\right) - \gamma \right) \,
17 17-gonal

(17, 17)

{17}

\scriptstyle \frac{x \, (14x+1)}{(1-x)^3} 17 \scriptstyle 15 \, (n-1) + 1

\scriptstyle 15n - 14

\scriptstyle \frac{1}{6} \, m \, (m+1) (15m-12)


\scriptstyle \frac{1}{2} \, m \, (m+1) (5m-4)

\scriptstyle \frac{2}{13} \left( \psi\left( m + \tfrac{2}{15} \right) - \psi(m+1) - \psi\left(\tfrac{2}{15}\right) - \gamma \right)  - \frac{ 2 \big( \psi\big( \tfrac{2}{15} \big) + \gamma \big) }{13}
18 18-gonal

(18, 18)

{18}

\scriptstyle \frac{x \, (15x+1)}{(1-x)^3} 18 \scriptstyle 16 \, (n-1) + 1

\scriptstyle 16n - 15

\scriptstyle \frac{1}{6} \, m \, (m+1) (16m-13) \scriptstyle \frac{1}{7} \left( \psi\left( m + \tfrac{1}{8} \right) - \psi(m+1) - \psi\left(\tfrac{1}{8}\right) - \gamma \right) \,
19 19-gonal

(19, 19)

{19}

\scriptstyle \frac{x \, (16x+1)}{(1-x)^3} 19 \scriptstyle 17 \, (n-1) + 1

\scriptstyle 17n - 16

\scriptstyle \frac{1}{6} \, m \, (m+1) (1 m-14) \scriptstyle \frac{2}{15} \left( \psi\left( m + \tfrac{2}{17} \right) - \psi(m+1 )- \psi\left(\tfrac{2}{17}\right) - \gamma \right)  - \frac{ 2 \left( \psi\left( \tfrac{2}{17} \right)+ \gamma \right) }{15}
20 20-gonal

(20, 20)

{20}

\scriptstyle \frac{x \, (17x+1)}{(1-x)^3} 20 \scriptstyle 18 \, (n-1) + 1

\scriptstyle 18n - 17

\scriptstyle \frac{1}{6} \, m \, (m+1) (18m-15)


\scriptstyle \frac{1}{2} \, m \, (m+1) (6m-5)

\scriptstyle \frac{1}{8} \left( \psi\left( m + \tfrac{1}{9} \right) - \psi(m+1) - \psi\left(\tfrac{1}{9}\right) - \gamma \right) \,
21 21-gonal

(21, 21)

{21}

\scriptstyle \frac{x \, (18x+1)}{(1-x)^3} 21 \scriptstyle 19 \, (n-1) + 1

\scriptstyle 19n - 18

\scriptstyle \frac{1}{6} \, m \, (m+1) (19m-16) \scriptstyle \frac{2}{17} \left( \psi\left( m + \tfrac{2}{19} \right) - \psi(m+1) - \psi\left(\tfrac{2}{19}\right) - \gamma \right)  - \frac{ 2 \left( \psi\left( \tfrac{2}{19} \right) + \gamma \right) }{17}
22 22-gonal

(22, 22)

{22}

\scriptstyle \frac{x \, (19x+1)}{(1-x)^3} 22 \scriptstyle 20 \, (n-1) + 1

\scriptstyle 20n - 19

\scriptstyle \frac{1}{6} \, m \, (m+1) (20m-17) \scriptstyle \frac{1}{9} \left( \psi\left( m + \tfrac{1}{10} \right) - \psi(m+1) - \psi\left(\tfrac{1}{10}\right) - \gamma \right) \,
23 23-gonal

(23, 23)

{23}

\scriptstyle \frac{x \, (20x+1)}{(1-x)^3} 23 \scriptstyle 21 \, (n-1) + 1

\scriptstyle 21n - 20

\scriptstyle \frac{1}{6} \, m \, (m+1) (21m-18)


\scriptstyle \frac{1}{2} \, m \, (m+1) (7m-6)

\scriptstyle \frac{2}{19} \left( \psi\left( m + \tfrac{2}{21} \right) - \psi(m+1) - \psi\left(\tfrac{2}{21}\right) - \gamma \right)  - \frac{ 2 \left( \psi\left( \tfrac{2}{21} \right)+ \gamma \right) }{19}
24 24-gonal

(24, 24)

{24}

\scriptstyle \frac{x \, (21x+1)}{(1-x)^3} 24\, \scriptstyle 22 \, (n-1) + 1

\scriptstyle 22n - 21

\scriptstyle \frac{1}{6} \, m \, (m+1) (22m-19) \scriptstyle \frac{1}{10} \left( \psi\left( m + \tfrac{1}{11} \right) - \psi(m+1) - \psi\left(\tfrac{1}{11}\right) - \gamma \right) \,
25 25-gonal

(25, 25)

{25}

\scriptstyle \frac{x \, (22x+1)}{(1-x)^3} 25 \scriptstyle 23 \, (n-1) + 1

\scriptstyle 23n - 22

\scriptstyle \frac{1}{6} \, m \, (m+1) (23m-20) \scriptstyle \frac{2}{21} \left( \psi\left( m + \tfrac{2}{23} \right) - \psi(m+1) - \psi\left(\tfrac{2}{23}\right) - \gamma \right)  - \frac{ 2 \left( \psi\left( \tfrac{2}{23} \right) + \gamma \right) }{21}
26 26-gonal

(26, 26)

{26}

\scriptstyle \frac{x \, (23x+1)}{(1-x)^3} 26 \scriptstyle 24 \, (n-1) + 1

\scriptstyle 24n - 23

\scriptstyle \frac{1}{6} \, m \, (m+1) (24m-21)


\scriptstyle \frac{1}{2} \, m \, (m+1) (8m-7)

\scriptstyle \frac{1}{11} \left( \psi\left( m + \tfrac{1}{12} \right) - \psi(m+1) - \psi\left(\tfrac{1}{12}\right) - \gamma \right) \,
27 27-gonal

(27, 27)

{27}

\scriptstyle \frac{x \, (24x+1)}{(1-x)^3} 27 \scriptstyle 25 \, (n-1) + 1

\scriptstyle 25n - 24

\scriptstyle \frac{1}{6} \, m \, (m+1) (25m-22) \scriptstyle \frac{2}{23} \left( \psi\left( m + \tfrac{2}{25} \right) - \psi(m+1) - \psi\left(\tfrac{2}{25}\right) - \gamma \right)  - \frac{ 2 \left(\psi\left( \tfrac{2}{25} \right) + \gamma \right) }{23}
28 28-gonal

(28, 28)

{28}

\scriptstyle \frac{x \, (25x+1)}{(1-x)^3} 28 \scriptstyle 26 \, (n-1) + 1

\scriptstyle 26n - 25

\scriptstyle \frac{1}{6} \, m \, (m+1) (26m-23) \scriptstyle \frac{1}{12} \left( \psi\left( m + \tfrac{1}{13} \right) - \psi(m+1) - \psi\left(\tfrac{1}{13}\right) - \gamma \right) \,
29 29-gonal

(29, 29)

{29}

\scriptstyle \frac{x \, (26x+1)}{(1-x)^3} 29 \scriptstyle 27 \, (n-1) + 1

\scriptstyle 27n - 26

\scriptstyle \frac{1}{6} \, m \, (m+1) (27m - 24)\,


\scriptstyle \frac{1}{2} \, m \, (m+1) (9m - 8)

\scriptstyle \frac{2}{25} \left( \psi\left( m + \tfrac{2}{27} \right) - \psi(m+1) - \psi\left(\tfrac{2}{27}\right) - \gamma \right)  - \frac{ 2 \big( \psi\big( \tfrac{2}{27} \big) + \gamma \big) }{25}
30 30-gonal

(30, 30)

{30}

\scriptstyle \frac{x \, (27x+1)}{(1-x)^3} 30 \scriptstyle 28 \, (n-1) + 1

\scriptstyle 28n - 27

\scriptstyle \frac{1}{6} \, m \, (m+1) (28m - 25) \scriptstyle \frac{1}{13} \left( \psi\left( m + \tfrac{1}{14} \right) - \psi(m+1) - \psi\left(\tfrac{1}{14}\right) - \gamma \right) \,

Table of sequences

For OEIS sequence numbers, refer to table of formulae and values above.

Polygonal numbers sequences
N0 P^{(2)}_{N_0}(n),\ n \ge 0 sequences 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, ...} A??????
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, ...} A??????
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, ...} A??????
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, ...} A??????
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, ...} A??????
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, ...}
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, ...} A??????
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, ...} A??????
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, ...} A??????
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, ...} A??????
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, ...} A??????
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, ...} A??????
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, ...} A??????
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, ...} A??????
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, ...} A??????
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, ...} A??????
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, ...} A??????
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, ...} A??????
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, ...} A??????
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, ...} A??????
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, ...} A??????
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, ...} A??????
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, ...} A??????
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, ...} A??????
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, ...} A??????
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, ...} A??????
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, ...} A??????
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, ...} A??????

See also

Notes

  1. Author of the plots: Stefan Friedrich Birkner, License: Creative Commons Attribution-ShareAlike 3.0 Unported.
  2. Where \scriptstyle P^{(d)}_{N_0}(n)\, is the d-dimensional regular convex polytope number with \scriptstyle N_0\, 0-dimensional facets, i.e. vertices V.
  3. Weisstein, Eric W., Polyhedral Formula, From MathWorld--A Wolfram Web Resource.
  4. 4.0 4.1 4.2 Weisstein, Eric W., Lagrange's Four-Square Theorem, From MathWorld--A Wolfram Web Resource.
  5. Weisstein, Eric W., Waring's Problem, From MathWorld--A Wolfram Web Resource.
  6. Weisstein, Eric W., Fifteen Theorem, From MathWorld--A Wolfram Web Resource.
  7. Where \scriptstyle 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, \scriptstyle N_0 = [(k+2)+(d-2)]\, is the number of vertices (including the \scriptstyle d-2\, apex vertices) of the polygonal base (hyper)pyramid.
  8. Sondow, Jonathan and Weisstein, Eric W., Harmonic Number, From MathWorld--A Wolfram Web Resource.
  9. Weisstein, Eric W., Euler-Mascheroni Constant, From MathWorld--A Wolfram Web Resource.
  10. Weisstein, Eric W., Digamma Function, From MathWorld--A Wolfram Web Resource.
  11. Weisstein, Eric W., Polygamma Function, From MathWorld--A Wolfram Web Resource.
  12. Weisstein, Eric W., Relatively Prime, From MathWorld--A Wolfram Web Resource.
  13. Weisstein, Eric W., Schläfli Symbol, From MathWorld--A Wolfram Web Resource.
  14. Downey, Lawrence M., Ong, Boon W., and Sellers, James A., Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, 2008.
  15. PSYCHEDELIC GEOMETRY, INVERSE POLYGONAL NUMBERS SERIES
  16. 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.
  17. Sondow, Jonathan and Weisstein, Eric W., Riemann Zeta Function, From MathWorld--A Wolfram Web Resource.

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