Pyramidal numbers

The pyramidal numbers are a family of sequences of 3-dimensional nonregular polytope numbers (among the 3-dimensional figurate numbers) formed by adding the first [N₀ - 1] positive polygonal numbers with constant number of sides [N₀ - 1], where N₀ is the number of vertices (including the apex vertex) of the pyramid of polygons. The term pyramid numbers is often used to refer to the square pyramidal numbers, having a polygonal base with four sides. The pyramidal numbers are a generalization of the pyramid numbers where the base is a regular convex polygon with any number of sides [N₀ - 1] ≥ 3. Triangular pyramidal numbers are known as tetrahedral numbers (one of the 5 regular polyhedral numbers, known as Platonic numbers and also one of the simplicial polytopic numbers). Pyramidal numbers may also be generalized to higher dimensions as hyperpyramidal numbers.

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

Formulae

The n^th [N₀-1]-gonal base pyramidal (having N₀ vertices) number is given by the formula:

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

$={\frac {1}{3}}{\binom {n+1}{2}}\{([N_{0}-1]-2)\,n-([N_{0}-1]-5)\}={\frac {n\,(n+1)}{6}}\{([N_{0}-1]-2)\,n-([N_{0}-1]-5)\},$

where

$P_{N_{0}}^{(2)}(n)=n+(N_{0}-2)\,P_{3}^{(2)}(n-1)=n+(N_{0}-2)\,T_{n-1}=n+(N_{0}-2){\binom {n}{2}}=n+(N_{0}-2){\frac {(n-1)\,n}{2}}={\frac {n}{2}}[(N_{0}-2)n-(N_{0}-4)],$

is the n^th polygonal number.

The roman geometers Epaphroditus and Vitrius Rufus (circa 150 AD) found the pyramidal number formula: (See User:Peter Luschny/FigurateNumber)

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

Descartes-Euler (convex) polyhedral formula

Descartes-Euler (convex) polyhedral formula:

${\sum _{i=0}^{2}(-1)^{i}N_{i}}=N_{0}-N_{1}+N_{2}=V-E+F=2,\,$

where N₀ is the number of 0-dimensional elements (vertices V,) N₁ is the number of 1-dimensional elements (edges E) and N₂ is the number of 2-dimensional elements (faces F) of the polyhedron.

Recurrence relation

$Y_{N_{0}}^{(3)}(n)=4Y_{N_{0}}^{(3)}(n-1)-6Y_{N_{0}}^{(3)}(n-2)+4Y_{N_{0}}^{(3)}(n-3)-Y_{N_{0}}^{(3)}(n-4)\,$

with initial conditions

$Y_{N_{0}}^{(3)}(0)=0\,$

$Y_{N_{0}}^{(3)}(1)=1\,$

$Y_{N_{0}}^{(3)}(2)=[N_{0}-1]+1\,$

$Y_{N_{0}}^{(3)}(3)=4[N_{0}-1]-2\,$

Generating function

$G_{\{Y_{N_{0}}^{(3)}(n)\}}(x)=x{{(([N_{0}-1]-3)x+1)} \over {(1-x)^{4}}}\,$

Order of basis

$g_{\{Y_{N_{0}}^{(3)}\}}=?$

In 1638, Fermat proposed that every positive integer is a sum of at most three triangular numbers, four square numbers, five pentagonal numbers, and k k-gonal numbers. Fermat claimed to have a proof of this result, although Fermat's proof has never been found. Joseph Louis Lagrange proved the square case (known as the four squares theorem) in 1770 and Gauss proved the triangular case in 1796. In 1813, Cauchy finally proved the horizontal generalization that every nonnegative integer can be written as a sum of k k-gonal numbers (known as the polygonal number theorem), while a vertical (higher dimensional) generalization has also been made (known as the Hilbert–Waring problem).

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

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

Differences

$Y_{N_{0}}^{(3)}(n)-Y_{N_{0}}^{(3)}(n-1)\,=P_{[N_{0}-1]}^{(2)}(n)$

Partial sums

$\sum _{n=1}^{m}{Y_{N_{0}}^{(3)}(n)}={\frac {1}{24}}m(m+1)(m+2)(([N_{0}-1]-2)m-[N_{0}-1]+6)={\frac {1}{4}}{\binom {m+2}{3}}(([N_{0}-1]-2)m-[N_{0}-1]+6)\,$

Partial sums of reciprocals

$\sum _{n=1}^{m}{1 \over {Y_{N_{0}}^{(3)}(n)}}=...\,$

Sum of reciprocals

$\sum _{n=1}^{\infty }{1 \over {Y_{N_{0}}^{(3)}(n)}}=-{\frac {6\{[N_{0}-1]-5+([N_{0}-1]-2)(\psi \left({\frac {3}{[N_{0}-1]-2}}\right)+\gamma )\}}{([N_{0}-1]-5)(2[N_{0}-1]-7)}},\,$

where $\scriptstyle \gamma \,$ is the Euler-Mascheroni constant and $\scriptstyle \psi (x)\,$ is the digamma function.

Table of formulae and values

Pyramidal numbers (A080851) obtained from pyramids of constructible polygons (with straightedge and compass) (A003401) are named in bold.

Pyramidal numbers formulae and values

N0−1	Name	Formulae $Y_{N_{0}}^{(3)}(n)=\,$ $\scriptstyle {\frac {T_{n}\{([{N_{0}}-1]-2)n-([{N_{0}}-1]-5)\}}{3}}$	n = 0	1	2	3	4	5	6	7	8	9	10	11	12	A-number
3	Triangular pyramidal	${\frac {T_{n}(n+2)}{3}}$ ${\frac {n(n+1)(n+2)}{6}}$ ${\frac {n^{(3)}}{3!}}$ ${\binom {n+2}{3}}$	0	1	4	10	20	35	56	84	120	165	220	286	364	A000292
4	Square pyramidal	${\frac {T_{n}(2n+1)}{3}}$ ${\frac {n(n+1)(2n+1)}{6}}$ ${\frac {(2n)^{(3)}}{4!}}$ ${\frac {1}{4}}{\binom {2n+2}{3}}$	0	1	5	14	30	55	91	140	204	285	385	506	650	A000330
5	Pentagonal pyramidal	${\frac {T_{n}(3n-0)}{3}}$ ${\frac {n^{2}(n+1)}{2}}$	0	1	6	18	40	75	126	196	288	405	550	726	936	A002411
6	Hexagonal pyramidal	${\frac {T_{n}(4n-1)}{3}}$	0	1	7	22	50	95	161	252	372	525	715	946	1222	A002412
7	Heptagonal pyramidal	${\frac {T_{n}(5n-2)}{3}}$	0	1	8	26	60	115	196	308	456	645	880	1166	1508	A002413
8	Octagonal pyramidal	${\frac {T_{n}(6n-3)}{3}}$	0	1	9	30	70	135	231	364	540	765	1045	1386	1794	A002414
9	9-gonal pyramidal	${\frac {T_{n}(7n-4)}{3}}$	0	1	10	34	80	155	266	420	624	885	1210	1606	2080	A007584
10	10-gonal pyramidal	${\frac {T_{n}(8n-5)}{3}}$	0	1	11	38	90	175	301	476	708	1005	1375	1826	2366	A007585
11	11-gonal pyramidal	${\frac {T_{n}(9n-6)}{3}}$	0	1	12	42	100	195	336	532	792	1125	1540	2046	2652	A007586
12	12-gonal pyramidal	${\frac {T_{n}(10n-7)}{3}}$	0	1	13	46	110	215	371	588	876	1245	1705	2266	2938	A007587
13	13-gonal pyramidal	${\frac {T_{n}(11n-8)}{3}}$	0	1	14	50	120	235	406	644	960	1365	1870	2486	3224	A050441
14	14-gonal pyramidal	${\frac {T_{n}(12n-9)}{3}}$	0	1	15
15	15-gonal pyramidal	${\frac {T_{n}(13n-10)}{3}}$	0	1	16
16	16-gonal pyramidal	${\frac {T_{n}(14n-11)}{3}}$	0	1	17
17	17-gonal pyramidal	${\frac {T_{n}(15n-12)}{3}}$	0	1	18
18	18-gonal pyramidal	${\frac {T_{n}(16n-13)}{3}}$	0	1	19
19	19-gonal pyramidal	${\frac {T_{n}(17n-14)}{3}}$	0	1	20
20	20-gonal pyramidal	${\frac {T_{n}(18n-15)}{3}}$	0	1	21
21	21-gonal pyramidal	${\frac {T_{n}(19n-16)}{3}}$	0	1	22
22	22-gonal pyramidal	${\frac {T_{n}(20n-17)}{3}}$	0	1	23
23	23-gonal pyramidal	${\frac {T_{n}(21n-18)}{3}}$	0	1	24
24	24-gonal pyramidal	${\frac {T_{n}(22n-19)}{3}}$	0	1	25
25	25-gonal pyramidal	${\frac {T_{n}(23n-20)}{3}}$	0	1	26
26	26-gonal pyramidal	${\frac {T_{n}(24n-21)}{3}}$	0	1	27
27	27-gonal pyramidal	${\frac {T_{n}(25n-22)}{3}}$	0	1	28
28	28-gonal pyramidal	${\frac {T_{n}(26n-23)}{3}}$	0	1	29
29	29-gonal pyramidal	${\frac {T_{n}(27n-24)}{3}}$	0	1	30
30	30-gonal pyramidal	${\frac {T_{n}(28n-25)}{3}}$	0	1	31

Table of related formulae and values

Pyramidal numbers (A080851) obtained from pyramids of constructible polygons (with straightedge and compass) (A003401) have the number of sides of their polygonal base shown in bold.

Pyramidal numbers related formulae and values

N0−1	Generating function $G_{\{Y_{N_{0}}^{(3)}(n)\}}(x)=$ $\scriptstyle {\frac {x\,(([N_{0}-1]-3)\,x+1)}{(1-x)^{4}}}$	Order of basis $g_{\{Y_{N_{0}}^{(3)}\}}=$	Differences $Y_{N_{0}}^{(3)}(n)-$ $Y_{N_{0}}^{(3)}(n-1)=$ $P_{[N_{0}-1]}^{(2)}(n)$	Partial sums $\sum _{n=1}^{m}Y_{N_{0}}^{(3)}(n)$	Partial sums of reciprocals $\sum _{n=1}^{m}{\frac {1}{Y_{N_{0}}^{(3)}(n)}}$	Sum of Reciprocals $\sum _{n=1}^{\infty }{\frac {1}{Y_{N_{0}}^{(3)}(n)}}$
3	${\frac {x}{(1-x)^{4}}}$	$\,$	$P_{3}^{(2)}(n)$	$\,$	${\frac {3{\binom {m+3}{3}}-m-3}{2{\binom {m+3}{3}}}}=$ ${\frac {3(m^{2}+3m)}{2(m^{2}+3m+2)}}$	${\frac {3}{2}}$ [1]
4	${\frac {x\,(x+1)}{(1-x)^{4}}}$	$\,$	$P_{4}^{(2)}(n)$	$\,$	$\,$	$6\,(3-4\log(2))$
5	${\frac {x\,(2x+1)}{(1-x)^{4}}}$	$\,$	$P_{5}^{(2)}(n)$	$\,$	$\,$	${\frac {\pi ^{2}}{3}}-2$
6	${\frac {x\,(3x+1)}{(1-x)^{4}}}$	$\,$	$P_{6}^{(2)}(n)$	$\,$	$\,$	${\frac {6}{5}}\,(12\log(2)-2\pi -1)$
7	${\frac {x\,(4x+1)}{(1-x)^{4}}}$	$\,$	$P_{7}^{(2)}(n)$	$\,$	$\,$	$\,$
8	${\frac {x\,(5x+1)}{(1-x)^{4}}}$	$\,$	$P_{8}^{(2)}(n)$	$\,$	$\,$	${\frac {2}{3}}\,(4\log(2)-1)$
9	${\frac {x\,(6x+1)}{(1-x)^{4}}}$	$\,$	$P_{9}^{(2)}(n)$	$\,$	$\,$	$\,$
10	${\frac {x\,(7x+1)}{(1-x)^{4}}}$	$\,$	$P_{10}^{(2)}(n)$	$\,$	$\,$	$\,$
11	${\frac {x\,(8x+1)}{(1-x)^{4}}}$	$\,$	$P_{11}^{(2)}(n)$	$\,$	$\,$	${\frac {9\log(3)+{\sqrt {3}}\,\pi -4}{10}}$
12	${\frac {x\,(9x+1)}{(1-x)^{4}}}$	$\,$	$P_{12}^{(2)}(n)$	$\,$	$\,$	$\,$
13	${\frac {x\,(10x+1)}{(1-x)^{4}}}$	$\,$	$P_{13}^{(2)}(n)$	$\,$	$\,$	$\,$
14	${\frac {x\,(11x+1)}{(1-x)^{4}}}$	$\,$	$P_{14}^{(2)}(n)$	$\,$	$\,$	$\,$
15	${\frac {x\,(12x+1)}{(1-x)^{4}}}$	$\,$	$P_{15}^{(2)}(n)$	$\,$	$\,$	$\,$
16	${\frac {x\,(13x+1)}{(1-x)^{4}}}$	$\,$	$P_{16}^{(2)}(n)$	$\,$	$\,$	$\,$
17	${\frac {x\,(14x+1)}{(1-x)^{4}}}$	$\,$	$P_{17}^{(2)}(n)$	$\,$	$\,$	$\,$
18	${\frac {x\,(15x+1)}{(1-x)^{4}}}$	$\,$	$P_{18}^{(2)}(n)$	$\,$	$\,$	$\,$
19	${\frac {x\,(16x+1)}{(1-x)^{4}}}$	$\,$	$P_{19}^{(2)}(n)$	$\,$	$\,$	$\,$
20	${\frac {x\,(17x+1)}{(1-x)^{4}}}$	$\,$	$P_{20}^{(2)}(n)$	$\,$	$\,$	$\,$
21	${\frac {x\,(18x+1)}{(1-x)^{4}}}$	$\,$	$P_{21}^{(2)}(n)$	$\,$	$\,$	$\,$
22	${\frac {x\,(19x+1)}{(1-x)^{4}}}$	$\,$	$P_{22}^{(2)}(n)\,$	$\,$	$\,$	$\,$
23	${\frac {x\,(20x+1)}{(1-x)^{4}}}$	$\,$	$P_{23}^{(2)}(n)$	$\,$	$\,$	$\,$
24	${\frac {x\,(21x+1)}{(1-x)^{4}}}$	$\,$	$P_{24}^{(2)}(n)$	$\,$	$\,$	$\,$
25	${\frac {x\,(22x+1)}{(1-x)^{4}}}$	$\,$	$P_{25}^{(2)}(n)\,$	$\,$	$\,$	$\,$
26	${\frac {x\,(23x+1)}{(1-x)^{4}}}$	$\,$	$P_{26}^{(2)}(n)$	$\,$	$\,$	$\,$
27	${\frac {x\,(24x+1)}{(1-x)^{4}}}$	$\,$	$P_{27}^{(2)}(n)$	$\,$	$\,$	$\,$
28	${\frac {x\,(25x+1)}{(1-x)^{4}}}$	$\,$	$P_{28}^{(2)}(n)\,$	$\,$	$\,$	$\,$
29	${\frac {x\,(26x+1)}{(1-x)^{4}}}$	$\,$	$P_{29}^{(2)}(n)$	$\,$	$\,$	$\,$
30	${\frac {x\,(27x+1)}{(1-x)^{4}}}$	$\,$	$P_{30}^{(2)}(n)$	$\,$	$\,$	$\,$

Table of sequences

$N_{0}-1$ is the number of vertices of the polygonal base of the pyramid ($N_{0}$ includes the vertex at the apex of the pyramid).

Pyramidal numbers sequences

$N_{0}-1$	$Y_{N_{0}}^{(3)}(n),\,n\geq 0$ sequences	A-number
3	{0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, 1140, 1330, 1540, 1771, 2024, 2300, 2600, 2925, 3276, 3654, 4060, 4495, ...}	A000292
4	{0, 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, 1015, 1240, 1496, 1785, 2109, 2470, 2870, 3311, 3795, 4324, 4900, 5525, 6201, 6930, 7714, ...}	A000330
5	{0, 1, 6, 18, 40, 75, 126, 196, 288, 405, 550, 726, 936, 1183, 1470, 1800, 2176, 2601, 3078, 3610, 4200, 4851, 5566, 6348, 7200, 8125, 9126, 10206, ...}	A002411
6	{0, 1, 7, 22, 50, 95, 161, 252, 372, 525, 715, 946, 1222, 1547, 1925, 2360, 2856, 3417, 4047, 4750, 5530, 6391, 7337, 8372, 9500, 10725, 12051, 13482, ...}	A002412
7	{0, 1, 8, 26, 60, 115, 196, 308, 456, 645, 880, 1166, 1508, 1911, 2380, 2920, 3536, 4233, 5016, 5890, 6860, 7931, 9108, 10396, 11800, 13325, 14976, ...}	A002413
8	{0, 1, 9, 30, 70, 135, 231, 364, 540, 765, 1045, 1386, 1794, 2275, 2835, 3480, 4216, 5049, 5985, 7030, 8190, 9471, 10879, 12420, 14100, 15925, 17901, ...}	A002414
9	{0, 1, 10, 34, 80, 155, 266, 420, 624, 885, 1210, 1606, 2080, 2639, 3290, 4040, 4896, 5865, 6954, 8170, 9520, 11011, 12650, 14444, 16400, 18525, ...}	A007584
10	{0, 1, 11, 38, 90, 175, 301, 476, 708, 1005, 1375, 1826, 2366, 3003, 3745, 4600, 5576, 6681, 7923, 9310, 10850, 12551, 14421, 16468, 18700, 21125, ...}	A007585
11	{0, 1, 12, 42, 100, 195, 336, 532, 792, 1125, 1540, 2046, 2652, 3367, 4200, 5160, 6256, 7497, 8892, 10450, 12180, 14091, 16192, 18492, 21000, 23725, ...}	A007586
12	{0, 1, 13, 46, 110, 215, 371, 588, 876, 1245, 1705, 2266, 2938, 3731, 4655, 5720, 6936, 8313, 9861, 11590, 13510, 15631, 17963, 20516, 23300, 26325, ...}	A007587
13	{0, 1, 14, 50, 120, 235, 406, 644, 960, 1365, 1870, 2486, 3224, 4095, 5110, 6280, 7616, 9129, 10830, 12730, 14840, 17171, 19734, 22540, 25600, 28925, ...}	A050441
14	{0, 1, 15, 54, 130, 255, 441, 700, 1044, 1485, 2035, 2706, 3510, 4459, 5565, 6840, 8296, 9945, 11799, 13870, 16170, 18711, 21505, 24564, 27900, 31525, ...}	A172073
15	{0, 1, 16, 58, 140, 275, 476, 756, 1128, 1605, 2200, 2926, 3796, 4823, 6020, 7400, 8976, 10761, 12768, 15010, 17500, 20251, 23276, 26588, 30200, ...}	A177890
16	{0, 1, 17, 62, 150, 295, 511, 812, 1212, 1725, 2365, 3146, 4082, 5187, 6475, 7960, 9656, 11577, 13737, 16150, 18830, 21791, 25047, 28612, 32500, ...}	A172076
17	{0, 1, 18, 66, 160, 315, 546, 868, 1296, 1845, 2530, 3366, 4368, 5551, 6930, 8520, 10336, 12393, 14706, 17290, 20160, 23331, 26818, 30636, 34800, ...}	A237616
18	{0, 1, 19, 70, 170, 335, 581, 924, 1380, 1965, 2695, 3586, 4654, 5915, 7385, 9080, 11016, 13209, 15675, 18430, 21490, 24871, 28589, 32660, 37100, ...}	A172078
19	{0, 1, 20, 74, 180, 355, 616, 980, 1464, 2085, 2860, 3806, 4940, 6279, 7840, 9640, 11696, 14025, 16644, 19570, 22820, 26411, 30360, 34684, 39400, ...}	A237617
20	{0, 1, 21, 78, 190, 375, 651, 1036, 1548, 2205, 3025, 4026, 5226, 6643, 8295, 10200, 12376, 14841, 17613, 20710, 24150, 27951, 32131, 36708, 41700, ... }	A172082
21	{0, 1, 22, 82, 200, 395, 686, 1092, 1632, 2325, 3190, 4246, 5512, 7007, 8750, 10760, 13056, 15657, 18582, 21850, 25480, 29491, 33902, 38732, 44000, ...}	A237618
22	{0, 1, 23, 86, 210, 415, 721, 1148, 1716, 2445, 3355, 4466, 5798, 7371, 9205, 11320, 13736, 16473, 19551, 22990, 26810, 31031, 35673, 40756, 46300, ...}	A172117
23	{0, 1, 24, 90, 220, 435, 756, 1204, 1800, 2565, 3520, 4686, 6084, 7735, 9660, 11880, 14416, 17289, 20520, 24130, 28140, 32571, 37444, 42780, 48600, ...}	A??????
24	{0, 1, 25, 94, 230, 455, 791, 1260, 1884, 2685, 3685, 4906, 6370, 8099, 10115, 12440, 15096, 18105, 21489, 25270, 29470, 34111, 39215, 44804, 50900, ... }	A??????
25	{0, 1, 26, 98, 240, 475, 826, 1316, 1968, 2805, 3850, 5126, 6656, 8463, 10570, 13000, 15776, 18921, 22458, 26410, 30800, 35651, 40986, 46828, 53200, ... }	A??????
26	{0, 1, 27, 102, 250, 495, 861, 1372, 2052, 2925, 4015, 5346, 6942, 8827, 11025, 13560, 16456, 19737, 23427, 27550, 32130, 37191, 42757, 48852, 55500, ... }	A??????
27	{0, 1, 28, 106, 260, 515, 896, 1428, 2136, 3045, 4180, 5566, 7228, 9191, 11480, 14120, 17136, 20553, 24396, 28690, 33460, 38731, 44528, 50876, 57800, ... }	A??????
28	{0, 1, 29, 110, 270, 535, 931, 1484, 2220, 3165, 4345, 5786, 7514, 9555, 11935, 14680, 17816, 21369, 25365, 29830, 34790, 40271, 46299, 52900, 60100, ... }	A??????
29	{0, 1, 30, 114, 280, 555, 966, 1540, 2304, 3285, 4510, 6006, 7800, 9919, 12390, 15240, 18496, 22185, 26334, 30970, 36120, 41811, 48070, 54924, 62400, ... }	A??????
30	{0, 1, 31, 118, 290, 575, 1001, 1596, 2388, 3405, 4675, 6226, 8086, 10283, 12845, 15800, 19176, 23001, 27303, 32110, 37450, 43351, 49841, 56948, ... }	A??????

See also

• Centered pyramidal numbers, i.e. (Centered polygons) pyramidal numbers.

External links

• Weisstein, Eric W., Pyramidal Number, from MathWorld—A Wolfram Web Resource.
• 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, 2^nd ed., 1994.