Centered Platonic numbers

The centered Platonic numbers, corresponding to the 5 Platonic solids (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

Formula

The n^th centered Platonic N₂-hedral (having N₀ vertices) number is given by the formula:

$\,_{c}P_{N_{0}}^{(3)}(n)={\frac {(2n+1)(k_{N_{0}}n^{2}+k_{N_{0}}n+3)}{3}},\,$

where $k_{N_{0}}=\{1,2,3,5,15\}\,$ for $N_{0}=\{4,6,8,12,20\}\,$ respectively, and N₀ is the number of 0-dimensional elements (vertices V.)

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

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

with initial conditions

$\,_{c}P_{N_{0}}^{(3)}(0)=0,\ \,_{c}P_{N_{0}}^{(3)}(1)=1,\ \,_{c}P_{N_{0}}^{(3)}(2)=N_{0}+1,\,$ (except for A005904?) :$\,_{c}P_{N_{0}}^{(3)}(3)=?,\ \,_{c}P_{N_{0}}^{(3)}(4)=?\,$

Generating function

$G_{\{\,_{c}P_{N_{0}}^{(3)}(n)\}}(x)={\frac {(1+x)(1+2(k_{N_{0}}-1)x+x^{2})}{(1-x)^{4}}},\,$

where $k_{N_{0}}=\{1,2,3,5,15\}\,$ for $N_{0}=\{4,6,8,12,20\}\,$ respectively, and N₀ is the number of 0-dimensional 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 k-polygonal 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-gon 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-gon 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

$\,_{c}P_{N_{0}}^{(3)}(n)-\,_{c}P_{N_{0}}^{(3)}(n-1)=?\,$

Partial sums

$\sum _{n=0}^{m}\,_{c}P_{N_{0}}^{(3)}(n)={\frac {m(m+2)(k_{N_{0}}(m+1)^{2}+6)}{6}},\,$

where $k_{N_{0}}=\{1,2,3,5,15\}\,$ for $N_{0}=\{4,6,8,12,20\}\,$, and N₀ is the number of 0-dimensional elements (vertices V.)

Partial sums of reciprocals

$\sum _{n=0}^{m}{\frac {1}{\,_{c}P_{N_{0}}^{(3)}(n)}}=?\,$

Sum of reciprocals

$\sum _{n=0}^{\infty }{\frac {1}{\,_{c}P_{N_{0}}^{(3)}(n)}}=?\,$

Table of formulae and values

N₀, N₁ and N₂ are the number of vertices (0-dimensional), edges (1-dimensional) and faces (2-dimensional) respectively, where the faces are the actual facets. The centered Platonic numbers are listed by increasing number N₀ of vertices.

Centered platonic numbers formulae and values

N0	Name (N0, N1, N2) Schläfli symbol	Formulae $\,_{c}P_{N_{0}}^{(3)}(n)=\,$ ${\frac {(2n+1)(k_{N_{0}}n^{2}+k_{N_{0}}n+3)}{3}},\,$ $k_{N_{0}}=\{1,2,3,5,15\}\,$	n = 0	1	2	3	4	5	6	7	8	9	10	11	12	OEIS number
4	Centered tetrahedral (4, 6, 4) {3, 3}	${\frac {(2n+1)(n^{2}+n+3)}{3}}\,$	1	5	15	35	69	121	195	295	425	589	791	1035	1325	A005894
6	Centered octahedral (6, 12, 8) {3, 4}	${\frac {(2n+1)(2n^{2}+2n+3)}{3}}\,$	1	7	25	63	129	231	377	575	833	1159	1561	2047	2625	A001845
8	Centered cube (8, 12, 6) {4, 3}	${\frac {(2n+1)(3n^{2}+3n+3)}{3}}\,$ $(2n+1)(n^{2}+n+1)\,$ $n^{3}+(n+1)^{3}\,$	1	9	35	91	189	341	559	855	1241	1729	2331	3059	3925	A005898
12	Centered icosahedral (12, 30, 20) {3, 5}	${\frac {(2n+1)(5n^{2}+5n+3)}{3}}\,$	1	13	55	147	309	561	923	1415	2057	2869	3871	5083	6525	A005902
20	Centered dodecahedral (20, 30, 12) {5, 3}	${\frac {(2n+1)(15n^{2}+15n+3)}{3}}\,$ $(2n+1)(5n^{2}+5n+1)\,$	1	33	155	427	909	1661	2743	4215	6137	8569	11571	15203	19525	A005904

Table of related formulae and values

N₀, N₁ and N₂ are the number of vertices (0-dimensional), edges (1-dimensional) and faces (2-dimensional) respectively, where the faces are the actual facets. The centered Platonic numbers are listed by increasing number N₀ of vertices.

Centered platonic numbers related formulae and values

N0	Name (N0, N1, N2) Schläfli symbol	Generating function $G_{\{\,_{c}P_{N_{0}}^{(3)}(n)\}}(x)=\,$ $\scriptstyle {\frac {(1+x)(1+2(k_{N_{0}}-1)x+x^{2})}{(1-x)^{4}}},\,$ $\scriptstyle k_{N_{0}}=\{1,2,3,5,15\}\,$	Order of basis $g_{\{\,_{c}P_{N_{0}}^{(3)}\}}=\,$	Differences $\,_{c}P_{N_{0}}^{(3)}(n)-\,$ $\,_{c}P_{N_{0}}^{(3)}(n-1)=\,$	Partial sums $\sum _{n=0}^{m}{\,_{c}P_{N_{0}}^{(3)}(n)}=$ $\scriptstyle {\frac {m(m+2)(k_{N_{0}}(m+1)^{2}+6)}{6}},\,$ $\scriptstyle k_{N_{0}}=\{1,2,3,5,15\}\,$	Partial sums of reciprocals $\sum _{n=0}^{m}{1 \over {\,_{c}P_{N_{0}}^{(3)}(n)}}=$	Sum of Reciprocals $\sum _{n=0}^{\infty }{1 \over {\,_{c}P_{N_{0}}^{(3)}(n)}}=$
4	Centered tetrahedral (4, 6, 4) {3, 3}	${\frac {(1+x)(1+0x+x^{2})}{(1-x)^{4}}}\,$ ${\frac {1-x^{4}}{(1-x)^{5}}}\,$	$\,$	$\,$	$\scriptstyle {\frac {m(m+2)(m^{2}+2m+7)}{6}}\,$	$\,$	$\,$
6	Centered octahedral (6, 12, 8) {3, 4}	${\frac {(1+x)(1+2x+x^{2})}{(1-x)^{4}}}\,$ ${\frac {(1+x)^{3}}{(1-x)^{4}}}\,$	$\,$	$\,$	$\scriptstyle {\frac {m(m+2)(2m^{2}+4m+8)}{6}}\,$ $\scriptstyle {\frac {m(m+2)(m^{2}+2m+4)}{3}}\,$	$\,$	$\,$
8	Centered cube (8, 12, 6) {4, 3}	${\frac {(1+x)(1+4x+x^{2})}{(1-x)^{4}}}\,$	$\,$	$\,$	$\scriptstyle {\frac {m(m+2)(3m^{2}+6m+9)}{6}}\,$ $\scriptstyle {\frac {m(m+2)(m^{2}+2m+3)}{2}}\,$	$\,$	$\,$
12	Centered icosahedral (12, 30, 20) {3, 5}	${\frac {(1+x)(1+8x+x^{2})}{(1-x)^{4}}}\,$	$\,$	$\,$	$\scriptstyle {\frac {m(m+2)(5m^{2}+10m+11)}{6}}\,$	$\,$	$\,$
20	Centered dodecahedral (20, 30, 12) {5, 3}	${\frac {(1+x)(1+28x+x^{2})}{(1-x)^{4}}}\,$	$\,$	$\,$	$\scriptstyle {\frac {m(m+2)(15m^{2}+30m+21)}{6}}\,$ $\scriptstyle {\frac {m(m+2)(5m^{2}+10m+7)}{2}}\,$	$\,$	$\,$

Table of sequences

Centered Platonic numbers sequences

N0	$\,_{c}P_{N_{0}}^{(3)}(n),\ n\geq 0\,$ 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

Platonic numbers

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.