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Centered Platonic numbers

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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 nth 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

Formula

The nth centered Platonic N2-hedral (having N0 vertices) number is given by the formula:[2]

\,_cP^{(3)}_{N_0}(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:[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

\,_cP^{(3)}_{N_0}(n) = 4 \,_cP^{(3)}_{N_2}(n-1) - 6 \,_cP^{(3)}_{N_2}(n-2) + 4 \,_cP^{(3)}_{N_2}(n-3) - \,_cP^{(3)}_{N_2}(n-4)\,

with initial conditions

\,_cP^{(3)}_{N_0}(0) = 0,\ \,_cP^{(3)}_{N_0}(1) = 1,\ \,_cP^{(3)}_{N_0}(2) = N_0+1,\, (except for A005904?) :\,_cP^{(3)}_{N_0}(3) = ?,\ \,_cP^{(3)}_{N_0}(4) = ?\,

Generating function

G_{\{\,_cP^{(3)}_{N_0}(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.[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-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 \ge 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 \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

\,_cP^{(3)}_{N_0}(n) - \,_cP^{(3)}_{N_0}(n-1) = ?\,

Partial sums

\sum_{n=0}^m \,_cP^{(3)}_{N_0}(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}{\,_cP^{(3)}_{N_0}(n)} = ?\,

Sum of reciprocals

\sum_{n=0}^{\infty} \frac{1}{\,_cP^{(3)}_{N_0}(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[5]

Formulae

\,_cP^{(3)}_{N_0}(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[6]

(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[5]

Generating

function

G_{\{\,_cP^{(3)}_{N_0}(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_{\{\,_cP^{(3)}_{N_0}\}} =\,

Differences

\,_cP^{(3)}_{N_0}(n) - \,

\,_cP^{(3)}_{N_0}(n-1) =\,

Partial sums

\sum_{n=0}^m {\,_cP^{(3)}_{N_0}(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{\,_cP^{(3)}_{N_0}(n)}} =

Sum of Reciprocals[7][8]

\sum_{n=0}^\infty{1\over{\,_cP^{(3)}_{N_0}(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+2 m+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+4 m+8)}{6}\,


\scriptstyle \frac{m (m+2) (m^2+2 m+4)}{3}\,

\, \,
8 Centered cube[6]

(8, 12, 6)

{4, 3}

\frac{(1+x)(1+4x+x^2)}{(1-x)^4}\, \, \, \scriptstyle \frac{m (m+2) (3m^2+6 m+9)}{6}\,


\scriptstyle \frac{m (m+2) (m^2+2 m+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+10 m+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+30 m+21)}{6}\,


\scriptstyle \frac{m (m+2) (5m^2+10 m+7)}{2}\,

\, \,


Table of sequences

Centered Platonic numbers sequences
N0 \,_cP^{(3)}_{N_0}(n),\ n \ge 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

Notes

  1. Weisstein, Eric W., Platonic Solid, From MathWorld--A Wolfram Web Resource.
  2. Where \scriptstyle \,_cP^{(d)}(N_0, n)\, is the centered d-dimensional regular convex polytope number with N_0\, 0-dimensional elements (vertices V.)
  3. Weisstein, Eric W., Polyhedral Formula, From MathWorld--A Wolfram Web Resource.
  4. Weisstein, Eric W., Fermat's Polygonal Number Theorem, From MathWorld--A Wolfram Web Resource.
  5. 5.0 5.1 Weisstein, Eric W., Schläfli Symbol, From MathWorld--A Wolfram Web Resource.
  6. 6.0 6.1 Weisstein, Eric W., Centered Cube Number, From MathWorld--A Wolfram Web Resource.
  7. Downey, Lawrence M., Ong, Boon W., and Sellers, James A., Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, 2008.
  8. PSYCHEDELIC GEOMETRY, INVERSE POLYGONAL NUMBERS SERIES.

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