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Pyramidal numbers

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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.

Contents

Formulae

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

Y^{(3)}_{N_0}(n) := \sum_{i=0}^{n} P^{(2)}_{[N_0 - 1]}(i) = 
                            \frac{P^{(2)}_{3}(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^{(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) \frac{(n-1) \, n}{2} = 
                           {\frac{n}{2}}[(N_0 - 2) n - (N_0 - 4)],

is the nth polygonal number.[2]

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

Y^{(3)}_{N_0}(n) := \sum_{i=0}^{n} P^{(2)}_{[N_0 - 1]}(i) = 
                            \frac{n+1}{6} \{ n + 2 P^{(2)}_{[N_0 - 1]}(n) \} = 
                            \frac{P^{(2)}_{3}(n) + (n+1) P^{(2)}_{[N_0 - 1]}(n)}{3}

Descartes-Euler (convex) polyhedral formula

Descartes-Euler (convex) polyhedral formula:[3]

{\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^{(3)}_{N_0}(n) = 4Y^{(3)}_{N_0}(n-1)-6Y^{(3)}_{N_0}(n-2)+4Y^{(3)}_{N_0}(n-3)-Y^{(3)}_{N_0}(n-4) \,

with initial conditions

Y^{(3)}_{N_0}(0) = 0 \,
Y^{(3)}_{N_0}(1) = 1 \,
Y^{(3)}_{N_0}(2) = [N_0-1]+1 \,
Y^{(3)}_{N_0}(3) = 4[N_0-1]-2 \,

Generating function

G_{\{Y^{(3)}_{N_0}(n)\}}(x) = x { {(([N_0-1]-3)x+1)} \over {(1-x)^4} } \,

Order of basis

g_{ \{ Y^{(3)}_{N_0} \} } = ?

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.[4] 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 \ge 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 dth powers, i.e. the set \scriptstyle \{ n^d \,|\, n = 0, 1, 2, \ldots \} of \scriptstyle dth 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 \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.

Differences

Y^{(3)}_{N_0}(n) - Y^{(3)}_{N_0}(n-1)\, = P^{(2)}_{[N_0-1]}(n)

Partial sums

\sum_{n=1}^m {Y^{(3)}_{N_0}(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^{(3)}_{N_0}(n)}} = ... \,

Sum of reciprocals

\sum_{n=1}^\infty {1\over{Y^{(3)}_{N_0}(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[5] and \scriptstyle \psi(x) \, is the digamma function.[6] [7]

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^{(3)}_{N_0}(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[8] \frac{T_n (n+2)}{3}


\frac{n (n+1)(n+2)}{6}


\frac{n^{(3)}}{3!}[9]


\binom{n+2}{3}

0 1 4 10 20 35 56 84 120 165 220 286 364 A000292
4 Square pyramidal[10] \frac{T_n (2n+1)}{3}


\frac{n (n+1)(2n+1)}{6}


\frac{(2n)^{(3)}}{4!}[9]


\frac{1}{4} \binom{2n+2}{3}

0 1 5 14 30 55 91 140 204 285 385 506 650 A000330
5 Pentagonal pyramidal[11][12] \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[13] \frac{T_n (4n-1)}{3} 0 1 7 22 50 95 161 252 372 525 715 946 1222 A002412
7 Heptagonal pyramidal[14] \frac{T_n (5n-2)}{3} 0 1 8 26 60 115 196 308 456 645 880 1166 1508 A002413
8 Octagonal pyramidal[15] \frac{T_n (6n-3)}{3} 0 1 9 30 70 135 231 364 540 765 1045 1386 1794 A002414
9 9-gonal pyramidal[16] \frac{T_n (7n-4)}{3} 0 1 10 34 80 155 266 420 624 885 1210 1606 2080 A007584
10 10-gonal pyramidal[17] \frac{T_n (8n-5)}{3} 0 1 11 38 90 175 301 476 708 1005 1375 1826 2366 A007585
11 11-gonal pyramidal[18] \frac{T_n (9n-6)}{3} 0 1 12 42 100 195 336 532 792 1125 1540 2046 2652 A007586
12 12-gonal pyramidal[19] \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^{(3)}_{N_0}(n) \} }(x) =

\scriptstyle \frac{ x \, (([N_0 - 1] - 3) \, x + 1) }{(1-x)^4}

Order
of basis

g_{ \{ Y^{(3)}_{N_0} \} } =

Differences

Y^{(3)}_{N_0}(n) -

Y^{(3)}_{N_0}(n-1) =


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

Partial sums

\sum_{n=1}^{m} Y^{(3)}_{N_0}(n)

Partial sums of reciprocals

\sum_{n=1}^{m} \frac{1}{Y^{(3)}_{N_0}(n)}

Sum of Reciprocals[20][21]

\sum_{n=1}^{\infty} \frac{1}{Y^{(3)}_{N_0}(n)}

3 \frac{x}{(1-x)^4} \, P^{(2)}_{3}(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^{(2)}_{4}(n) \, \, 6 \, (3 - 4 \log(2))
5 \frac{x \, (2x+1)}{(1-x)^4} \, P^{(2)}_{5}(n) \, \, \frac{\pi^2}{3} - 2
6 \frac{x \, (3x+1)}{(1-x)^4} \, P^{(2)}_{6}(n) \, \, \frac{6}{5} \, (12 \log(2) - 2 \pi - 1)
7 \frac{x \, (4x+1)}{(1-x)^4} \, P^{(2)}_{7}(n) \, \, \,
8 \frac{x \, (5x+1)}{(1-x)^4} \, P^{(2)}_{8}(n) \, \, \frac{2}{3} \, (4 \log(2) - 1)
9 \frac{x \, (6x+1)}{(1-x)^4} \, P^{(2)}_{9}(n) \, \, \,
10 \frac{x \, (7x+1)}{(1-x)^4} \, P^{(2)}_{10}(n) \, \, \,
11 \frac{x \, (8x+1)}{(1-x)^4} \, P^{(2)}_{11}(n) \, \, \frac{9 \log(3) + \sqrt{3} \, \pi - 4}{10}
12 \frac{x \, (9x+1)}{(1-x)^4} \, P^{(2)}_{12}(n) \, \, \,
13 \frac{x \, (10x+1)}{(1-x)^4} \, P^{(2)}_{13}(n) \, \, \,
14 \frac{x \, (11x+1)}{(1-x)^4} \, P^{(2)}_{14}(n) \, \, \,
15 \frac{x \, (12x+1)}{(1-x)^4} \, P^{(2)}_{15}(n) \, \, \,
16 \frac{x \, (13x+1)} {(1-x)^4} \, P^{(2)}_{16}(n) \, \, \,
17 \frac{x \, (14x+1)}{(1-x)^4} \, P^{(2)}_{17}(n) \, \, \,
18 \frac{x \, (15x+1)}{(1-x)^4} \, P^{(2)}_{18}(n) \, \, \,
19 \frac{x \, (16x+1)}{(1-x)^4} \, P^{(2)}_{19}(n) \, \, \,
20 \frac{x \, (17x+1)}{(1-x)^4} \, P^{(2)}_{20}(n) \, \, \,
21 \frac{x \, (18x+1)}{(1-x)^4} \, P^{(2)}_{21}(n) \, \, \,
22 \frac{x \, (19x+1)}{(1-x)^4} \, P^{(2)}_{22}(n)\, \, \, \,
23 \frac{x \, (20x+1)}{(1-x)^4} \, P^{(2)}_{23}(n) \, \, \,
24 \frac{x \, (21x+1)}{(1-x)^4} \, P^{(2)}_{24}(n) \, \, \,
25 \frac{x \, (22x+1)}{(1-x)^4} \, P^{(2)}_{25}(n)\, \, \, \,
26 \frac{x \, (23x+1)}{(1-x)^4} \, P^{(2)}_{26}(n) \, \, \,
27 \frac{x \, (24x+1)}{(1-x)^4} \, P^{(2)}_{27}(n) \, \, \,
28 \frac{x \, (25x+1)}{(1-x)^4} \, P^{(2)}_{28}(n)\, \, \, \,
29 \frac{x \, (26x+1)}{(1-x)^4} \, P^{(2)}_{29}(n) \, \, \,
30 \frac{x \, (27x+1)}{(1-x)^4} \, P^{(2)}_{30}(n) \, \, \,

Table of sequences

N0 − 1 is the number of vertices of the polygonal base of the pyramid (N0 includes the vertex at the apex of the pyramid).

Pyramidal numbers sequences
N0 − 1 Y^{(3)}_{N_0}(n),\, n \ge 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

Notes

  1. 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.
  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. [http://mathworld.wolfram.com/PolyhedralFormula.html]
  4. Weisstein, Eric W., Fermat's Polygonal Number Theorem, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/FermatsPolygonalNumberTheorem.html]
  5. Weisstein, Eric W., Euler-Mascheroni Constant, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/Euler-MascheroniConstant.html]
  6. Weisstein, Eric W., Digamma Function, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/DigammaFunction.html]
  7. Weisstein, Eric W., Polygamma Function, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/PolygammaFunction.html]
  8. Pyramid of triangular numbers.
  9. 9.0 9.1 Weisstein, Eric W., Rising Factorial, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/RisingFactorial.html]
  10. Pyramid of square numbers.
  11. Pyramid of pentagonal numbers.
  12. The row sums of the multiplication triangle yield pentagonal pyramidal numbers!
  13. Pyramid of hexagonal numbers.
  14. Pyramid of heptagonal numbers.
  15. Pyramid of octagonal numbers.
  16. Pyramid of 9-gonal numbers.
  17. Pyramid of 10-gonal numbers.
  18. Pyramid of 11-gonal numbers.
  19. Pyramid of 12-gonal numbers.
  20. Downey, Lawrence M., Ong, Boon W., and Sellers, James A., Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, 2008.
  21. PSYCHEDELIC GEOMETRY, INVERSE POLYGONAL NUMBERS SERIES.

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