A053016 -id:A053016 - cp's OEIS frontend

A359000 Number of undirected n-cycles of the octahedral graph.

Original entry on oeis.org

8, 15, 24, 16

Offset: 3

Seiichi Manyama, Dec 10 2022

a(3) = 8 = A053016(3).

a(6) = 16 = A268283(3)/2.

Eric Weisstein's World of Mathematics, Cycle Polynomial
Eric Weisstein's World of Mathematics, Octahedral Graph

Cf. A053016, A268283, A358999, A359001, A359002.

A359001 Number of undirected n-cycles of the dodecahedral graph.

Original entry on oeis.org

12, 0, 0, 30, 20, 36, 120, 100, 60, 180, 180, 90, 180, 130, 0, 30

Offset: 5

Seiichi Manyama, Dec 10 2022

a(5) = 12 = A053016(4).

a(20) = 30 = A268283(4)/2.

Eric Weisstein's World of Mathematics, Cycle Polynomial
Eric Weisstein's World of Mathematics, Dodecahedral Graph

Cf. A053016, A268283, A358999, A359000, A359002.

A359002 Number of undirected n-cycles of the icosahedral graph.

Original entry on oeis.org

20, 30, 72, 240, 720, 1620, 2680, 3336, 2880, 1280

Offset: 3

Seiichi Manyama, Dec 10 2022

a(3) = 20 = A053016(5).

a(12) = 1280 = A268283(5)/2.

Eric Weisstein's World of Mathematics, Cycle Polynomial
Eric Weisstein's World of Mathematics, Icosahedral Graph

Cf. A053016, A268283, A358999, A359000, A359001.

A201187 Number of distinct planar nets of the Platonic solids (in the order tetrahedron, cube, octahedron, dodecahedron, icosahedron).

Original entry on oeis.org

2, 11, 11, 43380, 43380

Offset: 1

Andrew McFarland, Nov 28 2011

A Platonic solid and its dual have the same number of nets.

			a(1) = 2 since a tetrahedron has 2 distinct nets.

Eric W. Weisstein, MathWorld: Net

Cf. A053016.

A299028 Number of vertices in the iterated clique graphs of the 1-skeleton of Plato's icosahedron.

Original entry on oeis.org

12, 20, 32, 92, 472

Offset: 0

José Hernández, Feb 01 2018

It is known that the sequence of orders of the iterated clique graphs of the icosahedron goes to infinity.

			By definition, the zeroth iterated clique graph of a graph G is equal to G itself; since the icosahedron has 12 vertices, a(0)=12.
The first iterated clique graph of the icosahedron has 20 vertices; hence, a(1)=20.

Miguel A. Pizaña, The icosahedron is clique divergent, Discrete Mathematics, 262 (Feb. 2003), pp. 229-239.
YAGS, YAGS - Yet Another Graph System

Cf. A299030, A063723, A053016.

K:=CliqueGraph;; g:=Icosahedron;; kg:=K(g);; Order(kg);
# It outputs the number of vertices in the first iterated
# clique graph of the icosahedron.

A299030 Number of vertices in the iterated clique graphs of the 1-skeleton of Plato's octahedron.

Original entry on oeis.org

6, 8, 16, 256, 340282366920938463463374607431768211456

Offset: 0

José Hernández, Feb 01 2018

The octahedron was the first known example of a k-divergent graph.

			By definition, the zeroth iterated clique graph of a graph G is equal to G itself; since the octahedron has 6 vertices, a(0)=6.
The first iterated clique graph of the octahedron has 8 vertices; hence, a(1)=8.

Miguel A. Pizaña, The icosahedron is clique divergent, Discrete Mathematics, 262 (Feb. 2003), pp. 229-239.
YAGS, YAGS - Yet Another Graph System

Cf. A299028, A063723, A053016.

Nest[Sqrt[2]^#&, 6, n] (* Omar Antolín-Camarena, May 16 2022 *)

K:=CliqueGraph;; g:=Octahedron;; kg:=K(g);; Order(kg);
# It outputs the number of vertices in the first iterated
# clique graph of the octahedron.

a(n) = sqrt(2)^sqrt(2)^...^sqrt(2)^6 with n occurrences of sqrt(2). - Omar Antolín-Camarena, May 16 2022

a(4) from Omar Antolín-Camarena, May 16 2022

A359202 Number of (bidimensional) faces of regular m-polytopes for m >= 3.

Original entry on oeis.org

4, 6, 8, 10, 12, 20, 24, 32, 35, 56, 80, 84, 96, 120, 160, 165, 220, 240, 280, 286, 364, 448, 455, 560, 672, 680, 720, 816, 960, 969, 1140, 1200, 1320, 1330, 1540, 1760, 1771, 1792, 2024, 2288, 2300, 2600, 2912, 2925, 3276, 3640, 3654, 4060, 4480, 4495, 4608

Offset: 1

Marco Ripà, Dec 20 2022

In 3 dimensions there are five (convex) regular polytopes and they have 4, 6, 8, 12, or 20 (bidimensional) faces (A053016).
In 4 dimensions there are six regular 4-polytopes and they have 10, 24, 32, 96, 720, or 1200 faces (A063925).
In m >= 5 dimensions, there are only 3 regular polytopes (i.e., the m-simplex, the m-cube, and the m-crosspolytope) so that we can sort their number of bidimensional faces in ascending order and define the present sequence.

			6 is a term since a cube has 6 faces.

Mathematics StackExchange, What are the formulas for the number of vertices, edges, faces, cells, 4-faces, ..., n-faces, of convex regular polytopes in n≥5 dimensions?
Wikipedia, List of regular polytopes and compounds

Cf. A000292, A001788, A053016, A063925, A130809.

Cf. A359201 (edges), A359662 (cells).

{a(n), n >= 1} = {{12, 96, 720, 1200} U {A000292} U {A001788} U {A130809}} \ {0, 1}.

A062554 Products of numbers of faces of Platonic solids, i.e., of {4,6,8,12,20}.

Original entry on oeis.org

4, 6, 8, 12, 16, 20, 24, 32, 36, 48, 64, 72, 80, 96, 120, 128, 144, 160, 192, 216, 240, 256, 288, 320, 384, 400, 432, 480, 512, 576, 640, 720, 768, 864, 960, 1024, 1152, 1280, 1296, 1440, 1536, 1600, 1728, 1920, 2048, 2304, 2400, 2560, 2592, 2880, 3072, 3200

Offset: 1

Neil Fernandez, Jul 02 2001

nonn

Cf. A053016, A062614.

ismult(n, v, vp) = {for (k=1, #v, q = n/v[k]; if ((type(q)== "t_INT") && vecsearch(vp, q), return (1)););}
lista(nn) = {v = [4,6,8,12,20]; vp = []; for (n=2, nn, if (vecsearch(v, n) || ismult(n, v, vp), print1(n, ", "); vp = concat(vp, n);););} \\ Michel Marcus, Feb 03 2015

More terms from Michel Marcus, Feb 03 2015

A140800 Total number of vertices in all finite n-dimensional convex regular polytopes, or 0 if the number is infinite.

Original entry on oeis.org

1, 2, 0, 50, 773, 48, 83, 150, 281, 540, 1055, 2082, 4133, 8232, 16427, 32814, 65585, 131124, 262199, 524346, 1048637, 2097216, 4194371, 8388678, 16777289, 33554508, 67108943, 134217810, 268435541, 536871000, 1073741915, 2147483742

Offset: 0

Jonathan Vos Post, Jul 15 2008

Andrew Weimholt suggests a related sequence, namely "total number of vertices in all finite n-dimensional regular polytopes, or 0 if the number is infinite, includes both convex and non-convex, beginning: 1, 2, 0, 106, 2453, 48, 83, 150, 281, 540, ... and writes that the sequence of just the non-convex cases (0, 0, -1, 56, 1680, 0, 0, 0, ..., where "-1" indicates infinity as zero is otherwise employed) is not as interesting, since it's all zeros from a(5) on.

			a(0) = 1 because the 0-D regular polytope is the point.
a(1) = 2 because the only regular 1-D polytope is the line segment, with 2 vertices, one at each end.
a(2) = 0, indicating infinity, because the regular k-gon has k vertices.
a(3) = 50 (4 for the tetrahedron, 6 for the octahedron, 8 for the cube, 12 for the icosahedron, 20 for the dodecahedron) = the sum of A053016.
a(4) = 773 = 5 + 8 + 16 + 24 + 120 + 600 = sum of A063924.
For n>4 there are only the three regular n-dimensional polytopes, the simplex with n+1 vertices, the hypercube with 2^n vertices and the hyperoctahedron = cross polytope = orthoplex with 2*n vertices, for a total of A086653(n) + 1 = 2^n + 3*n + 1 (again restricted to n > 4).

H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover, NY, 1973.
Branko Grunbaum, Convex Polytopes, second edition (first edition (1967) written with the cooperation of V. L. Klee, M. Perles and G. C. Shephard; second edition (2003) prepared by V. Kaibel, V. L. Klee and G. M. Ziegler), Graduate Texts in Mathematics, Vol. 221, Springer 2003.
P. McMullen and E. Schulte, Abstract Regular Polytopes, Encyclopedia of Mathematics and its Applications, Vol. 92, Cambridge University Press, Cambridge, 2002.

Georg Fischer, Table of n, a(n) for n = 0..1000
Gil Kalai, Five Open Problems Regarding Convex Polytopes.
Eric W. Weisstein, Polytope
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A000943, A000944, A019503, A053016, A060296, A063924, A063925, A063926, A063927, A065984, A086653, A093478, A093479, A105230, A105231.

LinearRecurrence[{4, -5, 2}, {1, 2, 0, 50, 773, 48, 83, 150}, 32] (* Georg Fischer, May 03 2019 *)

For n > 4, a(n) = A086653(n) + 1 = 2^n + 3*n + 1.

G.f.: -(1488*x^7 - 3656*x^6 + 2794*x^5 - 569*x^4 - 58*x^3 + 3*x^2 + 2*x - 1)/((1-x)^2*(1-2*x)). [Colin Barker, Sep 05 2012]

a(14)-a(15) corrected by Georg Fischer, May 02 2019

A146528 a(0) = 4; for n >= 1, a(n) = 2^n + 4.

Original entry on oeis.org

4, 6, 8, 12, 20, 36, 68, 132, 260, 516, 1028, 2052, 4100, 8196, 16388, 32772, 65540, 131076, 262148, 524292, 1048580, 2097156, 4194308, 8388612, 16777220, 33554436, 67108868, 134217732, 268435460, 536870916, 1073741828

Offset: 0

Roger L. Bagula, Oct 30 2008

This is one of many possible ways to extend the Platonic solids sequence A053016.

Cf. A053016, A342759.

Essentially the same as A140504.

v[n_] := 2*(If[n == 0, 0, 2^(n - 1)] + 2); Table[v[n], {n, 0, 30}]

a(n)=A140504(n), n>0. [From R. J. Mathar, Nov 05 2008]

Edited by N. J. A. Sloane, Mar 21 2021

cp's OEIS Frontend

A359000 Number of undirected n-cycles of the octahedral graph.

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A359001 Number of undirected n-cycles of the dodecahedral graph.

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A359002 Number of undirected n-cycles of the icosahedral graph.

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A201187 Number of distinct planar nets of the Platonic solids (in the order tetrahedron, cube, octahedron, dodecahedron, icosahedron).

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A299028 Number of vertices in the iterated clique graphs of the 1-skeleton of Plato's icosahedron.

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YAGS

A299030 Number of vertices in the iterated clique graphs of the 1-skeleton of Plato's octahedron.

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Mathematica

YAGS

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A359202 Number of (bidimensional) faces of regular m-polytopes for m >= 3.

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A062554 Products of numbers of faces of Platonic solids, i.e., of {4,6,8,12,20}.

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PARI

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A140800 Total number of vertices in all finite n-dimensional convex regular polytopes, or 0 if the number is infinite.

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