A063723 -id:A063723 - cp's OEIS frontend

A165953 Decimal expansion of (5sqrt(3) + sqrt(15))/(6Pi).

6, 6, 4, 9, 0, 8, 8, 9, 4, 2, 0, 5, 3, 2, 6, 6, 4, 3, 1, 1, 4, 4, 2, 8, 4, 4, 6, 7, 0, 8, 6, 3, 3, 7, 1, 6, 1, 6, 4, 8, 7, 6, 5, 8, 0, 5, 5, 5, 6, 9, 1, 9, 3, 8, 1, 0, 5, 7, 5, 9, 2, 6, 0, 5, 7, 2, 2, 9, 6, 4, 7, 1, 8, 1, 8, 7, 7, 3, 2, 5, 9, 7, 4, 9, 7, 0, 8, 9, 0, 0, 2, 6, 9, 2, 0, 9, 2, 5, 9, 8, 9, 8, 2, 8, 0

Offset: 0

Rick L. Shepherd, Oct 02 2009

The ratio of the volume of a regular dodecahedron to the volume of the circumscribed sphere (which has circumradius a*(sqrt(3) + sqrt(15))/4 = a*(A002194 + A010472)/4, where a is the dodecahedron's edge length; see MathWorld link). For similar ratios for other Platonic solids, see A165922, A049541, A165952, and A165954. A063723 shows the order of these by size.

			0.6649088942053266431144284467086337161648765805556919381057592605722964718...

Eric Weisstein's World of Mathematics, Dodecahedron.
Index entries for transcendental numbers.

Cf. A000796, A002194, A010472, A165922, A049541, A165952, A165954, A063723, A002163, A020760, A010527.

RealDigits[(5*Sqrt[3]+Sqrt[15])/(6*Pi),10,120][[1]] (* Harvey P. Dale, Feb 16 2018 *)

PARI
```
(5*sqrt(3)+sqrt(15))/(6*Pi)
```

Equals (5*A002194 + A010472)/(6*A000796).
Equals (5*A002194 + A010472)*A049541/6.
Equals (10*A010527 + A010472)*A049541/6.
Equals (5 + sqrt(5))/(2*Pi*sqrt(3)).
Equals (5 + A002163)*A049541*A020760/2.

A165954 Decimal expansion of sqrt(10 + 2sqrt(5))/(2Pi).

Original entry on oeis.org

6, 0, 5, 4, 6, 1, 3, 8, 2, 9, 1, 2, 5, 2, 5, 5, 8, 3, 3, 8, 6, 2, 6, 5, 2, 0, 5, 1, 2, 8, 0, 4, 4, 4, 9, 0, 3, 0, 0, 8, 4, 5, 4, 0, 8, 8, 0, 1, 4, 2, 8, 8, 9, 3, 3, 2, 0, 0, 9, 3, 5, 0, 0, 0, 8, 3, 8, 2, 9, 5, 6, 8, 3, 8, 2, 0, 7, 2, 7, 2, 7, 8, 5, 3, 6, 2, 4, 2, 6, 2, 5, 9, 6, 8, 8, 1, 3, 0, 5, 1, 9, 3, 2, 4, 1

Offset: 0

Rick L. Shepherd, Oct 04 2009

The ratio of the volume of a regular icosahedron to the volume of the circumscribed sphere (with circumradius a*sqrt(10 + 2*sqrt(5))/4 = a*A019881, where a is the icosahedron's edge length; see MathWorld link). For similar ratios for other Platonic solids, see A165922, A049541, A165952, and A165953. A063723 shows the order of these by size.

			0.6054613829125255833862652051280444903008454088014288933200935000838295683...

Eric Weisstein's World of Mathematics, Icosahedron.
Index entries for transcendental numbers.

Cf. A000796, A002163, A165922, A049541, A165952, A165953, A063723, A019881, A019694, A060294.

RealDigits[Sqrt[10+2Sqrt[5]]/(2Pi),10,120][[1]] (* Harvey P. Dale, Aug 27 2013 *)

PARI
```
sqrt(10+2*sqrt(5))/(2*Pi)
```

sqrt(10 + 2*sqrt(5))/(2*Pi) = sqrt(10 + 2*A002163)/(2*A000796) = 2*sin(2*Pi/5)/Pi = 2*sin(A019694)/A000796 = 2*sin(72 deg)/Pi = 2*A019881/A000796 = 2*A019881*A049541 = (2/Pi)*sin(72 deg) = A060294*A019881.

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

A234974 Expected lengths of random walks along the edges of a Platonic solid (in the order cube, octahedron, dodecahedron, icosahedron) from one vertex to an opposing one.

Original entry on oeis.org

10, 6, 35, 15

Offset: 1

Jens Voß, Jan 02 2014

For all Platonic solids (excluding the tetrahedron), the expected number of steps of a random walk from one vertex to its opposite vertex is indeed an integer.

Cf. comment to A063723

A244055 Number of edges on each face of the Platonic solids (in the order tetrahedron, cube, octahedron, dodecahedron, icosahedron).

Original entry on oeis.org

3, 4, 3, 5, 3

Offset: 1

Wesley Ivan Hurt, Jun 18 2014

The number of edges on the face of each Platonic solid is a divisor of the total number of edges (A063722) of its corresponding solid. The ratios of the total number of edges to face edges are 6:3, 12:4, 12:3, 30:5, 30:3 --> giving the integer sequence 2, 3, 4, 6, 10.
Although a(n) is also the number of vertices on each face of the Platonic solids, they are not altogether divisors of the total number of vertices (A063723) with the tetrahedron as the only exception. The ratios are 4:3, 8:4, 6:3, 20:5, 12:3.
Please see A053016 for an extensive list of web resources about the Platonic Solids.

Cf. A053016 (faces), A063722 (edges), A063723 (vertices).

A352622 Number of regular convex polytopes that can be formed with n indistinguishable points located at the vertices, coinciding in equal frequency at each vertex, if coinciding at all.

Original entry on oeis.org

1, 2, 2, 4, 3, 6, 3, 8, 4, 7, 3, 12, 3, 7, 6, 12, 3, 11, 3, 13, 6, 7, 3, 20, 5, 7, 6, 12, 3, 16, 3, 16, 6, 7, 7, 20, 3, 7, 6, 20, 3, 16, 3, 12, 10, 7, 3, 27, 5, 12, 6, 12, 3, 16, 7, 19, 6, 7, 3, 29, 3, 7, 10, 20, 7, 16, 3, 12, 6, 17, 3, 31, 3, 7, 10, 12, 7, 16

Offset: 1

Rajan Murthy, Mar 24 2022

nonn

For n = 1: there is only a 0-dimensional simplex.
For n = 2: the two points may coincide or may form a 1-dimensional simplex.
For n = 3: the three points may coincide or may form a 2-dimensional simplex.
For n = 2^(k+1), where k is a positive integer: a(n) = k + (k+2) + (k-1) + (k-1) = 4*k: k polygons (one for each factor > 2), k+2 simplexes (one for each factor), k-1 cubes (one for each even factor > 4, the cubes for 2 and 4 are a simplex and polygon, respectively), and k-1 orthoplexes (one for each even factor > 4, orthoplexes with 1, 2, and 4 vertices are already counted).
For prime numbers greater than 3 (n = p > 3, where p is prime): a(n) is always 3:
(1) the 0-dimensional polytope (all points coinciding), (2) a 2-dimensional p-gon, where p is a prime n, and (3) a (p-1)-dimensional simplex.
For even numbers which are not powers of 2: a(n) = 2*(number of factors) + (number of even factors) - 3 + adjustments. The adjustments are as follows: -1 if n is a multiple of 3; -1 if n is a multiple of 4; +1 for each positive integer k such that 2^(k+2) is a factor of n; +1 for each factor of n which is in the set (12,20,24,120,600). With the exception of factors 1 and 2, every factor contributes a simplex and a polygon. Even factors add a third polytope which is an orthoplex. Factors 1 and 2 only add a zero-dimensional and one-dimensional simplex respectively and so a total of three is subtracted (-1 for each of factors 1 + 2 and -1 for the even factor 2). The polygon and the simplex to which the factor of 3 maps are identical leading to an adjustment of -1. The polygon and the 2-dimensional "cube" that a factor of 4 maps to are identical also leading to a -1 adjustment. Factors which are powers of 2 greater than 4 and factors which correspond to a polytope peculiar to 3 or 4 dimensions each add one more possible polytope.
For nonprime odd numbers which are multiples of 3: a(n) = 2*(the number of factors) - 2. Each factor maps to a polygon and a simplex, but for the factor 3 the polygon is the simplex, and the factor 1 maps to a single coincident point.
For nonprime odd numbers which are not multiples of 3: a(n) = 2*(the number of factors) - 1. Each factor > 1 maps to a polygon and a simplex and the factor 1 maps to a single coincident point.

			For n = 12, the set of factors of 12 is (1, 2, 3, 4, 6, 12): 2 odd and 4 even including adjusting factors (3, 4, and 12). a(n) = 2*2 + 3*4 - 3 - 1 - 1 + 1 = 12: (1) a 0-dimensional simplex with 12 coincident points; (2) a 1-dimensional simplex with 2 groups of 6 coincident points; (3) a 2-dimensional simplex with 3 groups of 4 coincident points; (4,5) a square and a 3-dimensional simplex each with 4 groups of 3 coincident points; (6,7,8) a hexagon, an octahedron, and a 5-dimensional simplex each with 2 coincident points at the vertices; (9, 10, 11, 12) a dodecagon, a 6-dimensional orthoplex, an 11-dimensional simplex, and an icosahedron each with no coincident points.
For n = 20, the set of factors of 20 is (1, 2, 4, 5, 10, 20): 2 odd and 4 even including adjusting factors (4 and 20). a(n) = 2*2 + 3*4 - 3 - 1 + 1 = 13: (1) a 0-dimensional simplex with 20 coincident points; (2) a 1-dimensional simplex with 2 groups of 10 coincident points; (3, 4) a square and a 3-dimensional simplex each with 4 groups of 5 coincident points; (5, 6) a pentagon, and a 4-dimensional simplex each with groups of 4 coincident points; (7, 8, 9) a decagon, a 5-dimensional orthoplex, and a 9-dimensional simplex each with 2 coincident points at the vertices; (10, 11, 12, 13) a 20-sided polygon, a 10-dimensional orthoplex, a 19-dimensional simplex, and a dodecahedron.
For n = 24, the set of factors of 24 is (1, 2, 3, 4, 6, 8, 12, 24): 2 odd and 6 even including adjusting factors (3, 4, 8, 12, and 24). a(n) = 2*2 + 3*6 - 3 - 1 - 1 + 1 + 1 + 1 = 20: (1) a 0-dimensional simplex with 24 coincident points; (2) a 1-dimensional simplex with 2 groups of 12 coincident points; (3) a 2-dimensional simplex with 3 groups of 8 coincident points; (4, 5) a square and a 3-dimensional simplex each with 4 groups of 6 coincident points; (6, 7, 8) a hexagon, an octahedron, and a 5-dimensional simplex each with 4 coincident points; (9, 10, 11, 12) an octagon, a cube, a 4-dimensional orthoplex, a 7-dimensional simplex each with 3 coincident points; (13, 14, 15, 16) a dodecagon, a 6-dimensional orthoplex, an 11-dimensional simplex, and an icosahedron each with 2 coincident points; (17, 18, 19, 20) a 24-sided polygon, a 4-dimensional 24-cell, a 12-dimensional orthoplex, and a 23-dimensional simplex.

E. W. Weisstein, CRC Encyclopedia of Mathematics, 3rd Ed., CRC Press, 2009, 3037-3038.

Cf. A053016, A063723, A063927, A111336.

a(n) = Sum_{i|n} A111336(i).

cp's OEIS Frontend

A165953 Decimal expansion of (5*sqrt(3) + sqrt(15))/(6*Pi).

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PARI

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A165954 Decimal expansion of sqrt(10 + 2*sqrt(5))/(2*Pi).

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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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YAGS

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A234974 Expected lengths of random walks along the edges of a Platonic solid (in the order cube, octahedron, dodecahedron, icosahedron) from one vertex to an opposing one.

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A244055 Number of edges on each face of the Platonic solids (in the order tetrahedron, cube, octahedron, dodecahedron, icosahedron).

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A352622 Number of regular convex polytopes that can be formed with n indistinguishable points located at the vertices, coinciding in equal frequency at each vertex, if coinciding at all.

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A358960 Number of directed Hamiltonian paths of the Platonic graphs (in the order of tetrahedral, cubical, octahedral, dodecahedral, and icosahedral graph).

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A165953 Decimal expansion of (5sqrt(3) + sqrt(15))/(6Pi).

A165954 Decimal expansion of sqrt(10 + 2sqrt(5))/(2Pi).