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Pyramidal numbers
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The pyramidal numbers are a family of sequences of 3dimensional nonregular polytope numbers (among the 3dimensional figurate numbers) formed by adding the first [N_{0}  1] positive polygonal numbers with constant number of sides [N_{0}  1], where N_{0} 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 [N_{0}  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} [N_{0}1]gonal base pyramidal (having N_{0} vertices) number is given by the formula:^{[1]}
where
is the n^{th} polygonal number.^{[2]}
The roman geometers Epaphroditus and Vitrius Rufus (circa 150 AD) found the pyramidal number formula: (See User:Peter Luschny/FigurateNumber)
DescartesEuler (convex) polyhedral formula
DescartesEuler (convex) polyhedral formula:^{[3]}
where N_{0} is the number of 0dimensional elements (vertices V,) N_{1} is the number of 1dimensional elements (edges E) and N_{2} is the number of 2dimensional elements (faces F) of the polyhedron.
Recurrence relation
with initial conditions
Generating function
Order of basis
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 kgonal 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 kgonal 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 of nonnegative integers is called a basis of order if is the minimum number with the property that every nonnegative integer can be written as a sum of elements in . Lagrange’s sum of four squares can be restated as the set of nonnegative squares forms a basis of order 4.
Theorem (Cauchy). For every , the set of kgonal numbers forms a basis of order , i.e. every nonnegative integer can be written as a sum of kgon 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 such that every nonnegative integer is a sum of th powers, i.e. the set of th powers forms a basis of order . The HilbertWaring problem is concerned with the study of for . 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
Partial sums
Partial sums of reciprocals
Sum of reciprocals
where is the EulerMascheroni constant^{[5]} and 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.
N_{0}−1  Name  Formulae
 n = 0  1  2  3  4  5  6  7  8  9  10  11  12  Anumber 

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


3 
 [1]  
4  
5  
6  
7  
8  
9  
10  
11  
12  
13  
14  
15  
16  
17  
18  
19  
20  
21  
22  
23  
24  
25  
26  
27  
28  
29  
30 
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).
N_{0} − 1  sequences  Anumber 

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
 ↑ Where , k ≥ 1, n ≥ 0, is the ddimensional, d ≥ 0, (k+2)gonal base (hyper)pyramidal number where, for d ≥ 2, is the number of vertices (including the apex vertices) of the polygonal base [hyper]pyramid.
 ↑ Where is the ddimensional regular convex polytope number with 0dimensional facets, i.e. vertices V.
 ↑ Weisstein, Eric W., Polyhedral Formula, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/PolyhedralFormula.html]
 ↑ Weisstein, Eric W., Fermat's Polygonal Number Theorem, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/FermatsPolygonalNumberTheorem.html]
 ↑ Weisstein, Eric W., EulerMascheroni Constant, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/EulerMascheroniConstant.html]
 ↑ Weisstein, Eric W., Digamma Function, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/DigammaFunction.html]
 ↑ Weisstein, Eric W., Polygamma Function, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/PolygammaFunction.html]
 ↑ Pyramid of triangular numbers.
 ↑ ^{9.0} ^{9.1} Weisstein, Eric W., Rising Factorial, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/RisingFactorial.html]
 ↑ Pyramid of square numbers.
 ↑ Pyramid of pentagonal numbers.
 ↑ The row sums of the multiplication triangle yield pentagonal pyramidal numbers!
 ↑ Pyramid of hexagonal numbers.
 ↑ Pyramid of heptagonal numbers.
 ↑ Pyramid of octagonal numbers.
 ↑ Pyramid of 9gonal numbers.
 ↑ Pyramid of 10gonal numbers.
 ↑ Pyramid of 11gonal numbers.
 ↑ Pyramid of 12gonal numbers.
 ↑ 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.
External links
 Weisstein, Eric W., Pyramidal Number, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/PyramidalNumber.html]
 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.