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Centered Platonic numbers
The centered Platonic numbers, corresponding to the 5 Platonic solids^{[1]} (regular convex polyhedrons,) are defined by starting with 1 central dot (for n=0) and adding regular convex polyhedral layers around the central dot, where the n^{th} layer, n ≥ 1, has n+1 dots per facet ridge (face edge for polyhedrons) including both end vertices.
All figurate numbers are accessible via this structured menu: Classifications of figurate numbers
Contents
 1 Formula
 2 DescartesEuler (convex) polyhedral formula
 3 Recurrence relation
 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
Formula
The n^{th} centered Platonic N_{2}hedral (having N_{0} vertices) number is given by the formula:^{[2]}
where for respectively, and N_{0} is the number of 0dimensional elements (vertices V.)
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
 (except for A005904?) :
Generating function
where for respectively, and N_{0} is the number of 0dimensional elements (vertices V.)
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.^{[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 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
where for , and N_{0} is the number of 0dimensional elements (vertices V.)
Partial sums of reciprocals
Sum of reciprocals
Table of formulae and values
N_{0}, N_{1} and N_{2} are the number of vertices (0dimensional), edges (1dimensional) and faces (2dimensional) respectively, where the faces are the actual facets. The centered Platonic numbers are listed by increasing number N_{0} of vertices.
N_{0}  Name
(N_{0}, N_{1}, N_{2}) Schläfli symbol^{[5]} 
Formulae

n = 0  1  2  3  4  5  6  7  8  9  10  11  12  OEIS
number 

4  Centered tetrahedral
(4, 6, 4) {3, 3} 
1  5  15  35  69  121  195  295  425  589  791  1035  1325  A005894  
6  Centered octahedral
(6, 12, 8) {3, 4} 
1  7  25  63  129  231  377  575  833  1159  1561  2047  2625  A001845  
8  Centered cube^{[6]}
(8, 12, 6) {4, 3} 

1  9  35  91  189  341  559  855  1241  1729  2331  3059  3925  A005898 
12  Centered icosahedral
(12, 30, 20) {3, 5} 
1  13  55  147  309  561  923  1415  2057  2869  3871  5083  6525  A005902  
20  Centered dodecahedral
(20, 30, 12) {5, 3} 

1  33  155  427  909  1661  2743  4215  6137  8569  11571  15203  19525  A005904 
N_{0}, N_{1} and N_{2} are the number of vertices (0dimensional), edges (1dimensional) and faces (2dimensional) respectively, where the faces are the actual facets. The centered Platonic numbers are listed by increasing number N_{0} of vertices.
N_{0}  Name
(N_{0}, N_{1}, N_{2}) Schläfli symbol^{[5]} 
Generating
function

Order of
basis

Differences

Partial sums

Partial sums of reciprocals

Sum of Reciprocals^{[7]}^{[8]}


4  Centered tetrahedral
(4, 6, 4) {3, 3} 


6  Centered octahedral
(6, 12, 8) {3, 4} 



8  Centered cube^{[6]}
(8, 12, 6) {4, 3} 


12  Centered icosahedral
(12, 30, 20) {3, 5} 

20  Centered dodecahedral
(20, 30, 12) {5, 3} 

Table of sequences
N_{0}  sequences 

4  {1, 5, 15, 35, 69, 121, 195, 295, 425, 589, 791, 1035, 1325, 1665, 2059, 2511, 3025, 3605, 4255, 4979, 5781, 6665, 7635, 8695, 9849, 11101, 12455, 13915, 15485, ...} 
6  {1, 7, 25, 63, 129, 231, 377, 575, 833, 1159, 1561, 2047, 2625, 3303, 4089, 4991, 6017, 7175, 8473, 9919, 11521, 13287, 15225, 17343, 19649, 22151, 24857, 27775, ...} 
8  {1, 9, 35, 91, 189, 341, 559, 855, 1241, 1729, 2331, 3059, 3925, 4941, 6119, 7471, 9009, 10745, 12691, 14859, 17261, 19909, 22815, 25991, 29449, 33201, 37259, 41635, ...} 
12  {1, 13, 55, 147, 309, 561, 923, 1415, 2057, 2869, 3871, 5083, 6525, 8217, 10179, 12431, 14993, 17885, 21127, 24739, 28741, 33153, 37995, 43287, 49049, 55301, 62063, ...} 
20  {1, 33, 155, 427, 909, 1661, 2743, 4215, 6137, 8569, 11571, 15203, 19525, 24597, 30479, 37231, 44913, 53585, 63307, 74139, 86141, 99373, 113895, 129767, 147049, ...} 
See also
Notes
 ↑ Weisstein, Eric W., Platonic Solid, From MathWorldA Wolfram Web Resource.
 ↑ Where is the centered ddimensional regular convex polytope number with 0dimensional elements (vertices V.)
 ↑ Weisstein, Eric W., Polyhedral Formula, From MathWorldA Wolfram Web Resource.
 ↑ Weisstein, Eric W., Fermat's Polygonal Number Theorem, From MathWorldA Wolfram Web Resource.
 ↑ ^{5.0} ^{5.1} Weisstein, Eric W., Schläfli Symbol, From MathWorldA Wolfram Web Resource.
 ↑ ^{6.0} ^{6.1} Weisstein, Eric W., Centered Cube 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.