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 n^th 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

1 Formulae
2 Schläfli-Poincaré (convex) polytope formula
3 Recurrence equation
4 Generating function
5 Order of basis
6 Differences
7 Partial sums
8 Partial sums of reciprocals
9 Sum of reciprocals
10 Table of formulae and values
11 Table of related formulae and values
12 Table of sequences
13 See also
14 Notes
15 External links

Formulae

The n^th 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 n^th 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 n^th 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 n^th 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 m^th triangular number and $\scriptstyle Y^{(3)}_{N_0+1}(m)\,$ is the m^th 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 m^th 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 $x y$ 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 $P^{(2)}_{N_0}(n) = \,$ ${\frac{n}{2}[(N_0-2)n - (N_0-4)]} \,$	1	2	3	4	5	6	7	8	9	10	11	12	A-numbers
3	Triangular	$n(n+1)/2 \,$	1	3	6	10	15	21	28	36	45	55	66	78	A000217
4	Square	$n^2 \,$ $P^{(2)}_{3}(n-1) + P^{(2)}_{3}(n) \,$ $n + 2P^{(2)}_3(n-1) \,$	1	4	9	16	25	36	49	64	81	100	121	144	A000290
5	Pentagonal	$n(3n-1)/2 \,$	1	5	12	22	35	51	70	92	117	145	176	210	A000326
6	Hexagonal	$n(2n-1) \,$	1	6	15	28	45	66	91	120	153	190	231	276	A000384
7	Heptagonal	$n(5n-3)/2 \,$	1	7	18	34	55	81	112	148	189	235	286	342	A000566
8	Octagonal	$n(3n-2) \,$	1	8	21	40	65	96	133	176	225	280	341	408	A000567
9	9-gonal	$n(7n-5)/2 \,$	1	9	24	46	75	111	154	204	261	325	396	474	A001106
10	10-gonal	$n(4n-3) \,$	1	10	27	52	85	126	175	232	297	370	451	540	A001107
11	11-gonal	$n(9n-7)/2 \,$	1	11	30	58	95	141	196	260	333	415	506	606	A051682
12	12-gonal	$n(5n-4) \,$	1	12	33	64	105	156	217	288	369	460	561	672	A051624
13	13-gonal	$n(11n-9)/2 \,$	1	13	36	70	115	171	238	316	405	505	616	738	A051865
14	14-gonal	$n(6n-5) \,$	1	14	39	76	125	186	259	344	441	550	671	804	A051866
15	15-gonal	$n(13n-11)/2 \,$	1	15	42	82	135	201	280	372	477	595	726	870	A051867
16	16-gonal	$n(7n-6) \,$	1	16	45	88	145	216	301	400	513	640	781	936	A051868
17	17-gonal	$n(15n-13)/2 \,$	1	17	48	94	155	231	322	428	549	685	836	1002	A051869
18	18-gonal	$n(8n-7) \,$	1	18	51	100	165	246	343	456	585	730	891	1068	A051870
19	19-gonal	$n(17n-15)/2 \,$	1	19	54	106	175	261	364	484	621	775	946	1134	A051871
20	20-gonal	$n(9n-8) \,$	1	20	57	112	185	276	385	512	657	820	1001	1200	A051872
21	21-gonal	$n(19n-17)/2 \,$	1	21	60	118	195	291	406	540	693	865	1056	1266	A051873
22	22-gonal	$n(10n-9) \,$	1	22	63	124	205	306	427	568	729	910	1111	1332	A051874
23	23-gonal	$n(21n-19)/2 \,$	1	23	66	130	215	321	448	596	765	955	1166	1398	A051875
24	24-gonal	$n(11n-10) \,$	1	24	69	136	225	336	469	624	801	1000	1221	1464	A051876
25	25-gonal	$n(23n-21)/2 \,$	1	25	72	142	235	351	490	652	837	1045	1276	1530	A??????
26	26-gonal	$n(12n-11) \,$	1	26	75	148	245	366	511	680	873	1090	1331	1596	A??????
27	27-gonal	$n(25n-23)/2 \,$	1	27	78	154	255	381	532	708	909	1135	1386	1662	A??????
28	28-gonal	$n(13n-12) \,$	1	28	81	160	265	396	553	736	945	1180	1441	1728	A??????
29	29-gonal	$n(27n-25)/2 \,$	1	29	84	166	275	411	574	764	981	1225	1496	1794	A??????
30	30-gonal	$n(14n-13) \,$	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 $G_{\{P^{(2)}_{N_0}(n)\}}(x) =\,$ ${{x((N_0 - 3)x+1)}\over{(1-x)^3}}\,$	Order of basis $g_{\{P^{(2)}_{N_0}\}} = N_0,\,$ $N_0 \ge 3\,$ ^[4]	Differences $P^{(2)}_{N_0}(n) - \,$ $P^{(2)}_{N_0}(n-1) =\,$ $P^{(1)}_{N_0-1}(n) =\,$ $(N_0 - 2) (n-1) + 1\,$ Gnomonic numbers	Partial sums $\sum_{n=1}^m P^{(2)}_{N_0}(n) =\,$ $Y^{(3)}_{N_0+1}(m) =\,$ $\scriptstyle \frac{T_m [(N_0 - 2) m - (N_0 - 5)]}{3}\,$	Partial sums of reciprocals $\sum_{n=1}^m {1\over{P^{(2)}_{N_0}(n)}} =$ $\scriptstyle \frac{2 \big(\psi(m + \frac{2}{N_0-2}) -\psi(m+1) - \psi(\frac{2}{N_0-2}) - \gamma\big)}{(N_0-4)},\,$ $N_0 \neq 4.\,$	Sum of Reciprocals^[14]^[15] $\sum_{n=1}^\infty{1\over{P^{(2)}_{N_0}(n)}} =$ $- \frac{2\big(\psi\big(\frac{2}{N_0-2}\big)+\gamma\big)}{(N_0-4)},\,$ $N_0 \neq 4.\,$
3	Triangular (3, 3) {3}	$x\over{(1-x)^3}\,$	$3\,$	$1(n-1)+1 \,$ $n\,$	$\scriptstyle \frac{1}{6} m(m+1)(m+2)\,$	$\frac{2 m}{(1 + m)}\,$	$2 (\psi(2) + \gamma)\,$ $2\,$
4	Square (4, 4) {4}	${x(x+1)}\over{(1-x)^3} \,$	$4\,$	$2(n-1)+1 \,$ $2n-1\,$	$\scriptstyle \frac{1}{6} m(m+1)(2 m+1) \,$	$H_{m}^{(2)} \,$ ^[16]	${ {\zeta{(2)}} = {\pi^2\over6} }$ ^[17] Base 10: A013661
5	Pentagonal (5, 5) {5}	${x(2x+1)}\over{(1-x)^3}\,$	$5\,$	$3(n-1)+1\,$ $3n-2\,$	$\scriptstyle \frac{1}{6} m(m+1)(3 m-0)\,$ $\scriptstyle \frac{1}{2} m^2 (m+1)\,$	$\scriptstyle 2 \big(\psi(m+\tfrac{2}{3})-\psi(m+1)-\psi(\tfrac{2}{3})-\gamma\big)\,$	$- 2\big(\psi\big(\tfrac{2}{3}\big)+\gamma\big)\,$ ${ {3\ln\left(3\right)} - {\pi\sqrt{3}\over3} }$
6	Hexagonal (6, 6) {6}	${x(3x+1)}\over{(1-x)^3}\,$	$6\,$	$4(n-1)+1\,$ $4n-3\,$	$\scriptstyle \frac{1}{6} m(m+1)(4 m-1)\,$	$\scriptstyle \big(\psi(m+\tfrac{1}{2})-\psi(m+1)-\psi(\tfrac{1}{2})-\gamma\big)\,$	${ 2\ln\left(2\right) }$
7	Heptagonal (7, 7) {7}	${x(4x+1)}\over{(1-x)^3}\,$	$7\,$	$5(n-1)+1\,$ $5n-4\,$	$\scriptstyle \frac{1}{6} m (m+1) (5 m-2)\,$	$\scriptstyle \frac{2}{3} \big(\psi(m+\tfrac{2}{5})-\psi(m+1)-\psi(\tfrac{2}{5})-\gamma\big)\,$	$- \frac{2\big(\psi\big(\tfrac{2}{5}\big)+\gamma\big)}{3}\,$
8	Octagonal (8, 8) {8}	${x(5x+1)}\over{(1-x)^3}\,$	$8\,$	$6(n-1)+1\,$ $6n-5\,$	$\scriptstyle \frac{1}{6} m (m+1) (6 m-3)\,$ $\scriptstyle \frac{1}{2} m (m+1) (2 m-1)\,$	$\scriptstyle \frac{1}{2} \big(\psi(m+\tfrac{1}{3})-\psi(m+1)-\psi(\tfrac{1}{3})-\gamma\big)\,$	${ {3\ln\left(3\right)\over4} + {\pi\sqrt{3}\over12} }$
9	Nonagonal (9, 9) {9}	${x(6x+1)}\over{(1-x)^3}\,$	$9\,$	$7(n-1)+1\,$ $7n-6\,$	$\scriptstyle \frac{1}{6} m (m+1) (7 m-4)\,$	$\scriptstyle \frac{2}{5} \big(\psi(m+\tfrac{2}{7})-\psi(m+1)-\psi(\tfrac{2}{7})-\gamma\big)\,$	$- \frac{2\big(\psi\big(\tfrac{2}{7}\big)+\gamma\big)}{5}\,$
10	Decagonal (10, 10) {10}	${x(7x+1)}\over{(1-x)^3}\,$	$10\,$	$8(n-1)+1\,$ $8n-7\,$	$\scriptstyle \frac{1}{6} m (m+1) (8 m-5)\,$	$\scriptstyle \frac{1}{3} \big(\psi(m+\tfrac{1}{4})-\psi(m+1)-\psi(\tfrac{1}{4})-\gamma\big)\,$	${ {\ln\left(2\right)} + {\pi\over6} }$
11	Hendecagonal (11, 11) {11}	${x(8x+1)}\over{(1-x)^3}\,$	$11\,$	$9(n-1)+1\,$ $9n-8\,$	$\scriptstyle \frac{1}{6} m (m+1) (9 m-6)\,$ $\scriptstyle \frac{1}{2} m (m+1) (3 m-2)\,$	$\scriptstyle \frac{2}{7} \big(\psi(m+\tfrac{2}{9})-\psi(m+1)-\psi(\tfrac{2}{9})-\gamma\big)\,$	$- \frac{2\big(\psi\big(\tfrac{2}{9}\big)+\gamma\big)}{7}\,$
12	Dodecagonal (12, 12) {12}	${x(9x+1)}\over{(1-x)^3}\,$	$12\,$	$10(n-1)+1\,$ $10n-9\,$	$\scriptstyle \frac{1}{6} m (m+1) (10 m-7)\,$	$\scriptstyle \frac{1}{4} \big(\psi(m+\tfrac{1}{5})-\psi(m+1)-\psi(\tfrac{1}{5})-\gamma\big)\,$	$\,$
13	Tridecagonal (13, 13) {13}	${x(10x+1)}\over{(1-x)^3}\,$	$13\,$	$11(n-1)+1\,$ $11n-10\,$	$\scriptstyle \frac{1}{6} m (m+1) (11 m-8)\,$	$\scriptstyle \frac{2}{9} \big(\psi(m+\tfrac{2}{11})-\psi(m+1)-\psi(\tfrac{2}{11})-\gamma\big)\,$	$- \frac{2\big(\psi\big(\tfrac{2}{11}\big)+\gamma\big)}{9}\,$
14	Tetradecagonal (14, 14) {14}	${x(11x+1)}\over{(1-x)^3}\,$	$14\,$	$12(n-1)+1\,$ $12n-11\,$	$\scriptstyle \frac{1}{6} m (m+1) (12 m-9)\,$ $\scriptstyle \frac{1}{2} m (m+1) (4 m-3)\,$	$\scriptstyle \frac{1}{5} \big(\psi(m+\tfrac{1}{6})-\psi(m+1)-\psi(\tfrac{1}{6})-\gamma\big)\,$	${ {2\ln\left(2\right)\over5} + {3\ln\left(3\right)\over10} + {\pi\sqrt{3}\over10} }$
15	Pentadecagonal (15, 15) {15}	${x(12x+1)}\over{(1-x)^3}\,$	$15\,$	$13(n-1)+1\,$ $13n-12\,$	$\scriptstyle \frac{1}{6} m (m+1) (13 m-10)\,$	$\scriptstyle \frac{2}{11} \big(\psi(m+\tfrac{2}{13})-\psi(m+1)-\psi(\tfrac{2}{13})-\gamma\big)\,$	$- \frac{2\big(\psi\big(\tfrac{2}{13}\big)+\gamma\big)}{11}\,$
16	Hexadecagonal (16, 16) {16}	${x(13x+1)}\over{(1-x)^3}\,$	$16\,$	$14(n-1)+1\,$ $14n-13\,$	$\scriptstyle \frac{1}{6} m (m+1) (14 m-11)\,$	$\scriptstyle \frac{1}{6} \big(\psi(m+\tfrac{1}{7})-\psi(m+1)-\psi(\tfrac{1}{7})-\gamma\big)\,$	$\,$
17	Heptadecagonal (17, 17) {17}	${x(14x+1)}\over{(1-x)^3}\,$	$17\,$	$15(n-1)+1\,$ $15n-14\,$	$\scriptstyle \frac{1}{6} m (m+1) (15 m-12)\,$ $\scriptstyle \frac{1}{2} m (m+1) (5 m-4)\,$	$\scriptstyle \frac{2}{13} \big(\psi(m+\tfrac{2}{15})-\psi(m+1)-\psi(\tfrac{2}{15})-\gamma\big)\,$	$- \frac{2\big(\psi\big(\tfrac{2}{15}\big)+\gamma\big)}{13}\,$
18	Octadecagonal (18, 18) {18}	${x(15x+1)}\over{(1-x)^3}\,$	$18\,$	$16(n-1)+1\,$ $16n-15\,$	$\scriptstyle \frac{1}{6} m (m+1) (16 m-13)\,$	$\scriptstyle \frac{1}{7} \big(\psi(m+\tfrac{1}{8})-\psi(m+1)-\psi(\tfrac{1}{8})-\gamma\big)\,$	$\,$
19	Nonadecagonal (19, 19) {19}	${x(16x+1)}\over{(1-x)^3}\,$	$19\,$	$17(n-1)+1\,$ $17n-16\,$	$\scriptstyle \frac{1}{6} m (m+1) (17 m-14)\,$	$\scriptstyle \frac{2}{15} \big(\psi(m+\tfrac{2}{17})-\psi(m+1)-\psi(\tfrac{2}{17})-\gamma\big)\,$	$- \frac{2\big(\psi\big(\tfrac{2}{17}\big)+\gamma\big)}{15}\,$
20	Icosagonal (20, 20) {20}	${x(17x+1)}\over{(1-x)^3}\,$	$20\,$	$18(n-1)+1\,$ $18n-17\,$	$\scriptstyle \frac{1}{6} m (m+1) (18 m-15)\,$ $\scriptstyle \frac{1}{2} m (m+1) (6 m-5)\,$	$\scriptstyle \frac{1}{8} \big(\psi(m+\tfrac{1}{9})-\psi(m+1)-\psi(\tfrac{1}{9})-\gamma\big)\,$	$\,$
21	Icosihenagonal (21, 21) {21}	${x(18x+1)}\over{(1-x)^3}\,$	$21\,$	$19(n-1)+1\,$ $19n-18\,$	$\scriptstyle \frac{1}{6} m (m+1) (19 m-16)\,$	$\scriptstyle \frac{2}{17} \big(\psi(m+\tfrac{2}{19})-\psi(m+1)-\psi(\tfrac{2}{17})-\gamma\big)\,$	$- \frac{2\big(\psi\big(\tfrac{2}{19}\big)+\gamma\big)}{17}\,$
22	Icosidigonal (22, 22) {22}	${x(19x+1)}\over{(1-x)^3}\,$	$22\,$	$20(n-1)+1\,$ $20n-19\,$	$\scriptstyle \frac{1}{6} m (m+1) (20 m-17)\,$	$\scriptstyle \frac{1}{9} \big(\psi(m+\tfrac{1}{10})-\psi(m+1)-\psi(\tfrac{1}{10})-\gamma\big)\,$	$\,$
23	Icositrigonal (23, 23) {23}	${x(20x+1)}\over{(1-x)^3}\,$	$23\,$	$21(n-1)+1\,$ $21n-20\,$	$\scriptstyle \frac{1}{6} m (m+1) (21 m-18)\,$ $\scriptstyle \frac{1}{2} m (m+1) (7 m-6)\,$	$\scriptstyle \frac{2}{19} \big(\psi(m+\tfrac{2}{21})-\psi(m+1)-\psi(\tfrac{2}{21})-\gamma\big)\,$	$- \frac{2\big(\psi\big(\tfrac{2}{21}\big)+\gamma\big)}{19}\,$
24	Icositetragonal (24, 24) {24}	${x(21x+1)}\over{(1-x)^3}\,$	$24\,$	$22(n-1)+1\,$ $22n-21\,$	$\scriptstyle \frac{1}{6} m (m+1) (22 m-19)\,$	$\scriptstyle \frac{1}{10} \big(\psi(m+\tfrac{1}{11})-\psi(m+1)-\psi(\tfrac{1}{11})-\gamma\big)\,$	$\,$
25	Icosipentagonal (25, 25) {25}	${x(22x+1)}\over{(1-x)^3}\,$	$25\,$	$23(n-1)+1\,$ $23n-22\,$	$\scriptstyle \frac{1}{6} m (m+1) (23 m-20)\,$	$\scriptstyle \frac{2}{21} \big(\psi(m+\tfrac{2}{23})-\psi(m+1)-\psi(\tfrac{2}{23})-\gamma\big)\,$	$- \frac{2\big(\psi\big(\tfrac{2}{23}\big)+\gamma\big)}{21}\,$
26	Icosihexagonal (26, 26) {26}	${x(23x+1)}\over{(1-x)^3}\,$	$26\,$	$24(n-1)+1\,$ $24n-23\,$	$\scriptstyle \frac{1}{6} m (m+1) (24 m-21)\,$ $\scriptstyle \frac{1}{2} m (m+1) (8 m-7)\,$	$\scriptstyle \frac{1}{11} \big(\psi(m+\tfrac{1}{12})-\psi(m+1)-\psi(\tfrac{1}{12})-\gamma\big)\,$	$\,$
27	Icosiheptagonal (27, 27) {27}	${x(24x+1)}\over{(1-x)^3}\,$	$27\,$	$25(n-1)+1\,$ $25n-24\,$	$\scriptstyle \frac{1}{6} m (m+1) (25 m-22)\,$	$\scriptstyle \frac{2}{23} \big(\psi(m+\tfrac{2}{25})-\psi(m+1)-\psi(\tfrac{2}{25})-\gamma\big)\,$	$- \frac{2\big(\psi\big(\tfrac{2}{25}\big)+\gamma\big)}{23}\,$
28	Icosioctagonal (28, 28) {28}	${x(25x+1)}\over{(1-x)^3}\,$	$28\,$	$26(n-1)+1\,$ $26n-25\,$	$\scriptstyle \frac{1}{6} m (m+1) (26 m-23)\,$	$\scriptstyle \frac{1}{12} \big(\psi(m+\tfrac{1}{13})-\psi(m+1)-\psi(\tfrac{1}{13})-\gamma\big)\,$	$\,$
29	Icosinonagonal (29, 29) {29}	${x(26x+1)}\over{(1-x)^3}\,$	$29\,$	$27(n-1)+1\,$ $27n-26\,$	$\scriptstyle \frac{1}{6} m (m+1) (27 m-24)\,$ $\scriptstyle \frac{1}{2} m (m+1) (9 m-8)\,$	$\scriptstyle \frac{2}{25} \big(\psi(m+\tfrac{2}{27})-\psi(m+1)-\psi(\tfrac{2}{27})-\gamma\big)\,$	$- \frac{2\big(\psi\big(\tfrac{2}{27}\big)+\gamma\big)}{25}\,$
30	Triacontagonal (30, 30) {30}	${x(27x+1)}\over{(1-x)^3}\,$	$30\,$	$28(n-1)+1\,$ $28n-27\,$	$\scriptstyle \frac{1}{6} m (m+1) (28 m-25)\,$	$\scriptstyle \frac{1}{13} \big(\psi(m+\tfrac{1}{14})-\psi(m+1)-\psi(\tfrac{1}{14})-\gamma\big)\,$	$\,$

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
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, 630, 666, 703, 741, 780, 820, 861, 903, 946, ...}
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, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, ...}
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, 1426, 1520, 1617, 1717, 1820, 1926, 2035, 2147, ...}
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, 1891, 2016, 2145, 2278, 2415, 2556, 2701, 2850, ...}
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, 2205, 2356, 2512, 2673, 2839, 3010, 3186, 3367, ...}
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, 2640, 2821, 3008, 3201, 3400, 3605, 3816, ...}
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, 3075, 3286, 3504, 3729, 3961, 4200, 4446, ...}
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, 3277, 3510, 3751, 4000, 4257, 4522, 4795, 5076, ...}
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, 3683, 3945, 4216, 4496, 4785, 5083, 5390, 5706, ...}
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, 4089, 4380, 4681, 4992, 5313, 5644, 5985, ...}
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, 4495, 4815, 5146, 5488, 5841, 6205, 6580, ...}
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, 4901, 5250, 5611, 5984, 6369, 6766, 7175, ...}
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, 4942, 5307, 5685, 6076, 6480, 6897, 7327, 7770, ...}
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, 5320, 5713, 6120, 6541, 6976, 7425, 7888, 8365, ...}
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, 5698, 6119, 6555, 7006, 7472, 7953, 8449, 8960, ...}
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, 6076, 6525, 6990, 7471, 7968, 8481, 9010, ...}
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, 6454, 6931, 7425, 7936, 8464, 9009, 9571, ...}
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, 6832, 7337, 7860, 8401, 8960, 9537, 10132, ...}
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, 7210, 7743, 8295, 8866, 9456, 10065, 10693, ...}
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, 7588, 8149, 8730, 9331, 9952, 10593, 11254, ...}
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, 7966, 8555, 9165, 9796, 10448, 11121, 11815, ...}
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, 8344, 8961, 9600, 10261, 10944, 11649, ...}
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, 8722, 9367, 10035, 10726, 11440, 12177, ...}
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, 9100, 9773, 10470, 11191, 11936, 12705, ...}
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, 9478, 10179, 10905, 11656, 12432, 13233, ...}
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, 9856, 10585, 11340, 12121, 12928, 13761, ...}
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, 10234, 10991, 11775, 12586, 13424, 14289, ...}
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, 10612, 11397, 12210, 13051, 13920, 14817, ...}

Notes

↑ Author of the plots: Stefan Friedrich Birkner, License: Creative Commons Attribution-ShareAlike 3.0 Unported.
↑ 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.
↑ Weisstein, Eric W., Polyhedral Formula, From MathWorld--A Wolfram Web Resource.
↑ ^4.0 ^4.1 ^4.2 Weisstein, Eric W., Lagrange's Four-Square Theorem, From MathWorld--A Wolfram Web Resource.
↑ Weisstein, Eric W., Waring's Problem, From MathWorld--A Wolfram Web Resource.
↑ Weisstein, Eric W., Fifteen Theorem, From MathWorld--A Wolfram Web Resource.
↑ 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.
↑ Sondow, Jonathan and Weisstein, Eric W., Harmonic Number, From MathWorld--A Wolfram Web Resource.
↑ Weisstein, Eric W., Euler-Mascheroni Constant, From MathWorld--A Wolfram Web Resource.
↑ Weisstein, Eric W., Digamma Function, From MathWorld--A Wolfram Web Resource.
↑ Weisstein, Eric W., Polygamma Function, From MathWorld--A Wolfram Web Resource.
↑ Weisstein, Eric W., Relatively Prime, From MathWorld--A Wolfram Web Resource.
↑ Weisstein, Eric W., Schläfli Symbol, From MathWorld--A Wolfram Web Resource.
↑ Downey, Lawrence M., Ong, Boon W., and Sellers, James A., Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, 2008.
↑ PSYCHEDELIC GEOMETRY, INVERSE POLYGONAL NUMBERS SERIES
↑ 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.
↑ Sondow, Jonathan and Weisstein, Eric W., Riemann Zeta Function, From MathWorld--A Wolfram Web Resource.

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.

[0] Author of the plots: Stefan Friedrich Birkner, License: Creative Commons Attribution-ShareAlike 3.0 Unported.

[1] 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.

[2] Weisstein, Eric W., Polyhedral Formula, From MathWorld--A Wolfram Web Resource.

[FermatsPolygonalNumberTheorem-3] 4.0 ^4.1 ^4.2 Weisstein, Eric W., Lagrange's Four-Square Theorem, From MathWorld--A Wolfram Web Resource.

[4] Weisstein, Eric W., Waring's Problem, From MathWorld--A Wolfram Web Resource.

[5] Weisstein, Eric W., Fifteen Theorem, From MathWorld--A Wolfram Web Resource.

[Pyramidal_Numbers_Notation-6] 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.

[HarmonicNumber-7] Sondow, Jonathan and Weisstein, Eric W., Harmonic Number, From MathWorld--A Wolfram Web Resource.

[Euler-MascheroniConstant-8] Weisstein, Eric W., Euler-Mascheroni Constant, From MathWorld--A Wolfram Web Resource.

[DigammaFunction-9] Weisstein, Eric W., Digamma Function, From MathWorld--A Wolfram Web Resource.

[PolygammaFunction-10] Weisstein, Eric W., Polygamma Function, From MathWorld--A Wolfram Web Resource.

[11] Weisstein, Eric W., Relatively Prime, From MathWorld--A Wolfram Web Resource.

[Schlafli-12] Weisstein, Eric W., Schläfli Symbol, From MathWorld--A Wolfram Web Resource.

[13] Downey, Lawrence M., Ong, Boon W., and Sellers, James A., Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, 2008.

[14] PSYCHEDELIC GEOMETRY, INVERSE POLYGONAL NUMBERS SERIES

[GENERALIZED_HARMONIC_NUMBER_IDENTITIES-15] 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.

[16] Sondow, Jonathan and Weisstein, Eric W., Riemann Zeta Function, From MathWorld--A Wolfram Web Resource.

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Polygonal numbers

From OeisWiki

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