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(Centered polygons) pyramidal numbers
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The centered pyramidal numbers, i.e. (centered polygons) 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 centered polygonal numbers with constant number of sides [N_{0}1], where N_{0} is the number of vertices (including the apex vertex) of the polygonal base pyramid. The term centered pyramid numbers, i.e. (centered squares) pyramidal numbers, is often used to refer to the centered square pyramidal numbers, i.e. (centered squares) pyramidal numbers, having a polygonal base with four sides. The centered pyramidal numbers, i.e. (centered polygons) pyramidal numbers, are a generalization of the centered pyramid numbers, i.e. (centered squares) pyramidal numbers, where the base is a regular convex polygon with any number of sides [N_{0}1] ≥ 3. Centered pyramidal numbers, i.e. (centered polygons) pyramidal numbers, may also be generalized to higher dimensions as centered hyperpyramidal numbers, i.e. (centered polygons) hyperpyramidal numbers.
While the (centered polygons) pyramidal numbers are pyramidal stacks of centered polygons, the generated figures are NOT GLOBALLY CENTERED. Thus the (centered polygons) pyramidal numbers do not belong to the category of globally centered figurate numbers (which start with the globally central dot, giving value 1, for n = 0,) they belong to the category of globally noncentered figurate numbers (which equals 0 for n = 0 and start with the initial dot, giving value 1, for n = 1.) It would be less confusing if the centered pyramidal numbers were called (centered polygons) pyramidal numbers. Note that although the triangular pyramidal numbers are tetrahedral numbers, the (centered triangles) pyramidal numbers are NOT the (globally centered) centered tetrahedral numbers.
All figurate numbers are accessible via this structured menu: Classifications of figurate numbers
Formulae
The n^{th} [N_{0}1]gonal base centered pyramidal number is given by the formula: ^{[1]}
where is the n^{th} N_{0}sided centered polygonal number.
Observe that for [N_{0}1] = 6, the formula simplifies to that of the cubes which means that the n^{th} centered hexagonal pyramidal number may be rearranged as the n^{th} cube!
DescartesEuler (convex) polyhedral formula
DescartesEuler (convex) polyhedral formula:^{[2]}
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 kpolygonal numbers. Fermat claimed to have a proof of this result, although Fermat's proof has never been found.^{[3]} 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 kgon numbers (known as the polygonal number theorem,) while a vertical (higher dimensional) generalization has also been made (known as the HilbertWaring 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 kgon 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
Table of formulae and values
Table of related formulae and values
N_{0}−1  Generating
function
 Order
of basis
 Differences
 Partial sums
 Partial sums of reciprocals
 Sum of Reciprocals^{[6]}^{[7]}


3  
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  sequences 

3  {1, 5, 15, 34, 65, 111, 175, 260, 369, 505, 671, 870, 1105, 1379, 1695, 2056, 2465, 2925, 3439, 4010, 4641, 5335, 6095, 6924, 7825, 8801, 9855, 10990, 12209, 13515, 14911, ...} 
4  {1, 6, 19, 44, 85, 146, 231, 344, 489, 670, 891, 1156, 1469, 1834, 2255, 2736, 3281, 3894, 4579, 5340, 6181, 7106, 8119, 9224, 10425, 11726, 13131, 14644, 16269, 18010, ...} 
5  {1, 7, 23, 54, 105, 181, 287, 428, 609, 835, 1111, 1442, 1833, 2289, 2815, 3416, 4097, 4863, 5719, 6670, 7721, 8877, 10143, 11524, 13025, 14651, 16407, 18298, 20329, 22505, ...} 
6  {1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 13824, 15625, 17576, 19683, 21952, 24389, ...} 
7  {1, 9, 31, 74, 145, 251, 399, 596, 849, 1165, 1551, 2014, 2561, 3199, 3935, 4776, 5729, 6801, 7999, 9330, 10801, 12419, 14191, 16124, 18225, 20501, 22959, 25606, 28449, ...} 
8  {1, 10, 35, 84, 165, 286, 455, 680, 969, 1330, 1771, 2300, 2925, 3654, 4495, 5456, 6545, 7770, 9139, 10660, 12341, 14190, 16215, 18424, 20825, 23426, 26235, 29260, 32509, ...} 
9  {1, 11, 39, 94, 185, 321, 511, 764, 1089, 1495, 1991, 2586, 3289, 4109, 5055, 6136, 7361, 8739, 10279, 11990, 13881, 15961, 18239, 20724, 23425, 26351, 29511, 32914, ...} 
10  {1, 12, 43, 104, 205, 356, 567, 848, 1209, 1660, 2211, 2872, 3653, 4564, 5615, 6816, 8177, 9708, 11419, 13320, 15421, 17732, 20263, 23024, 26025, 29276, 32787, 36568, ...} 
11  {1, 13, 47, 114, 225, 391, 623, 932, 1329, 1825, 2431, 3158, 4017, 5019, 6175, 7496, 8993, 10677, 12559, 14650, 16961, 19503, 22287, 25324, 28625, 32201, 36063, 40222, ...} 
12  {1, 14, 51, 124, 245, 426, 679, 1016, 1449, 1990, 2651, 3444, 4381, 5474, 6735, 8176, 9809, 11646, 13699, 15980, 18501, 21274, 24311, 27624, 31225, 35126, 39339, 43876, ...} 
13  {1, 15, 55, 134, 265, 461, 735, 1100, 1569, 2155, 2871, 3730, 4745, 5929, 7295, 8856, 10625, 12615, 14839, 17310, 20041, 23045, 26335, 29924, 33825, 38051, 42615, 47530, ...} 
14  {1, 16, 59, 144, 285, 496, 791, 1184, 1689, 2320, 3091, 4016, 5109, 6384, 7855, 9536, 11441, 13584, 15979, 18640, 21581, 24816, 28359, 32224, 36425, 40976, 45891, 51184, ...} 
15  {1, 17, 63, 154, 305, 531, 847, 1268, 1809, 2485, 3311, 4302, 5473, 6839, 8415, 10216, 12257, 14553, 17119, 19970, 23121, 26587, 30383, 34524, 39025, 43901, 49167, 54838, ...} 
16  {1, 18, 67, 164, 325, 566, 903, 1352, 1929, 2650, 3531, 4588, 5837, 7294, 8975, 10896, 13073, 15522, 18259, 21300, 24661, 28358, 32407, 36824, 41625, 46826, 52443, ...} 
17  {1, 19, 71, 174, 345, 601, 959, 1436, 2049, 2815, 3751, 4874, 6201, 7749, 9535, 11576, 13889, 16491, 19399, 22630, 26201, 30129, 34431, 39124, 44225, 49751, 55719, ...} 
18  {1, 20, 75, 184, 365, 636, 1015, 1520, 2169, 2980, 3971, 5160, 6565, 8204, 10095, 12256, 14705, 17460, 20539, 23960, 27741, 31900, 36455, 41424, 46825, 52676, 58995, ...} 
19  {1, 21, 79, 194, 385, 671, 1071, 1604, 2289, 3145, 4191, 5446, 6929, 8659, 10655, 12936, 15521, 18429, 21679, 25290, 29281, 33671, 38479, 43724, 49425, 55601, 62271, ...} 
20  {1, 22, 83, 204, 405, 706, 1127, 1688, 2409, 3310, 4411, 5732, 7293, 9114, 11215, 13616, 16337, 19398, 22819, 26620, 30821, 35442, 40503, 46024, 52025, 58526, 65547, ...} 
21  {1, 23, 87, 214, 425, 741, 1183, 1772, 2529, 3475, 4631, 6018, 7657, 9569, 11775, 14296, 17153, 20367, 23959, 27950, 32361, 37213, 42527, 48324, 54625, 61451, 68823, ...} 
22  {1, 24, 91, 224, 445, 776, 1239, 1856, 2649, 3640, 4851, 6304, 8021, 10024, 12335, 14976, 17969, 21336, 25099, 29280, 33901, 38984, 44551, 50624, 57225, 64376, 72099, ...} 
23  {1, 25, 95, 234, 465, 811, 1295, 1940, 2769, 3805, 5071, 6590, 8385, 10479, 12895, 15656, 18785, 22305, 26239, 30610, 35441, 40755, 46575, 52924, 59825, 67301, 75375, ...} 
24  {1, 26, 99, 244, 485, 846, 1351, 2024, 2889, 3970, 5291, 6876, 8749, 10934, 13455, 16336, 19601, 23274, 27379, 31940, 36981, 42526, 48599, 55224, 62425, 70226, 78651, ...} 
25  {1, 27, 103, 254, 505, 881, 1407, 2108, 3009, 4135, 5511, 7162, 9113, 11389, 14015, 17016, 20417, 24243, 28519, 33270, 38521, 44297, 50623, 57524, 65025, 73151, 81927, ...} 
26  {1, 28, 107, 264, 525, 916, 1463, 2192, 3129, 4300, 5731, 7448, 9477, 11844, 14575, 17696, 21233, 25212, 29659, 34600, 40061, 46068, 52647, 59824, 67625, 76076, 85203, ...} 
27  {1, 29, 111, 274, 545, 951, 1519, 2276, 3249, 4465, 5951, 7734, 9841, 12299, 15135, 18376, 22049, 26181, 30799, 35930, 41601, 47839, 54671, 62124, 70225, 79001, 88479, ...} 
28  {1, 30, 115, 284, 565, 986, 1575, 2360, 3369, 4630, 6171, 8020, 10205, 12754, 15695, 19056, 22865, 27150, 31939, 37260, 43141, 49610, 56695, 64424, 72825, 81926, 91755, ...} 
29  {1, 31, 119, 294, 585, 1021, 1631, 2444, 3489, 4795, 6391, 8306, 10569, 13209, 16255, 19736, 23681, 28119, 33079, 38590, 44681, 51381, 58719, 66724, 75425, 84851, 95031, ...} 
30  {1, 32, 123, 304, 605, 1056, 1687, 2528, 3609, 4960, 6611, 8592, 10933, 13664, 16815, 20416, 24497, 29088, 34219, 39920, 46221, 53152, 60743, 69024, 78025, 87776, 98307, ...} 
See also
Notes
 ↑ Where , k ≥ 1, n ≥ 0, is the ddimensional, d ≥ 0, (k+2)gonal base (centered polygons) (hyper)pyramidal number where, for d ≥ 2, is the number of vertices (including the apex vertices) of the (centered polygonal base) (hyper)pyramid (the quoted emphasizes that only the polygons are centered, not the whole figure.)
 ↑ Weisstein, Eric W., Polyhedral Formula, From MathWorldA Wolfram Web Resource.
 ↑ Weisstein, Eric W., Fermat's Polygonal Number Theorem, From MathWorldA Wolfram Web Resource.
 ↑ Weisstein, Eric W., Stellated octahedron, From MathWorldA Wolfram Web Resource.
 ↑ Weisstein, Eric W., Stella Octangula Number, From MathWorldA 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.
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, 2^{nd} ed., 1994.