A298808 -id:A298808 - cp's OEIS frontend

A161435 Number of reduced words of length n in the Weyl group A_3 (or D_3).

1, 3, 5, 6, 5, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

John Cannon and N. J. A. Sloane, Nov 30 2009

a(n) is also the number of vertices of a truncated octahedron (the Voronoi cell for the lattice A_3*) at edge distance n from a given vertex. See also row 4 of the triangle in A008302. - N. J. A. Sloane, Oct 12 2015, corrected Aug 26 2016.
If the zeros are omitted, this is the coordination sequence for the truncated octahedron (see Karzes link). - N. J. A. Sloane, Jan 08 2020
Computed with Magma using commands similar to those used to compute A161409.

N. Bourbaki, Groupes et alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche I.)
J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.

Tom Karzes, Polyhedron Coordination Sequences
Index entries for growth series for groups

Cf. A008302, A161409.

Growth series for groups D_n, n = 3,...,50: A161435, A162207, A162208, A162209, A162210, A162211, A162212, A162248, A162288, A162297, A162300, A162301, A162321, A162327, A162328, A162346, A162347, A162359, A162360, A162364, A162365, A162366, A162367, A162368, A162369, A162370, A162376, A162377, A162378, A162379, A162380, A162381, A162384, A162388, A162389, A162392, A162399, A162402, A162403, A162411, A162412, A162413, A162418, A162452, A162456, A162461, A162469, A162492; also A162206.

# Growth series for D_k, truncated to terms of order M. - N. J. A. Sloane, Aug 07 2021
f := proc(m::integer) (1-x^m)/(1-x) ; end proc:
g := proc(k,M) local a,i; global f;
a:=f(k)*mul(f(2*i),i=1..k-1);
seriestolist(series(a,x,M+1));
end proc;

CoefficientList[Series[(1 - x^2) (1 - x^3) (1 - x^4) / (1 - x)^3, {x, 0, 20}], x] (* Vincenzo Librandi, Aug 23 2016 *)

G.f. for A_m is the polynomial Product_{k=1..m} (1-x^(k+1))/(1-x). Only finitely many terms are nonzero. This is a row of the triangle in A008302.

A161696 Number of reduced words of length n in the Weyl group B_3.

Original entry on oeis.org

1, 3, 5, 7, 8, 8, 7, 5, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

John Cannon and N. J. A. Sloane, Nov 30 2009

nonn

If the zeros are ignored, this is the coordination sequence for the truncated cuboctahedron (see the Karzes link). - N. J. A. Sloane, Jan 08 2020

Computed with MAGMA using commands similar to those used to compute A161409.

J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.
N. Bourbaki, Groupes et algèbres de Lie, Chap. 4, 5, 6. (The group is defined in Planche II.)

Tom Karzes, Polyhedron Coordination Sequences

The growth series for the finite Coxeter (or Weyl) groups B_2 through B_12 are A161696-A161699, A161716, A161717, A161733, A161755, A161776, A161858. These are all rows of A128084. The growth series for the affine Coxeter (or Weyl) groups B_2 through B_12 are A008576, A008137, A267167-A267175.

m:=10; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[1-t^(2*k): k in [1..3]])/(1-t)^3)); // G. C. Greubel, Oct 25 2018

seq(coeff(series(mul((1-x^(2k))/(1-x),k=1..3),x,n+1), x, n), n = 0 .. 120); # Muniru A Asiru, Oct 25 2018

CoefficientList[Series[Product[(1-x^(2*k)), {k,1,3}] /(1-x)^3, {x,0,9}], x] (* G. C. Greubel, Oct 25 2018 *)

t='t+O('t^10); Vec(prod(k=1,3,1-t^(2*k))/(1-t)^3) \\ G. C. Greubel, Oct 25 2018

G.f. for B_m is the polynomial Prod_{k=1..m}(1-x^(2k))/(1-x). Only finitely many terms are nonzero. This is a row of the triangle in A128084.

A329500 Coordination sequence for 1-skeleton of truncated icosahedron or "Buckyball".

Original entry on oeis.org

1, 3, 6, 8, 10, 10, 10, 8, 3, 1

Offset: 0

N. J. A. Sloane, Nov 17 2019

The truncated icosahedron is a polyhedron with 60 vertices and 32 faces. There are 20 hexagonal faces and 12 pentagonal faces. It is also known as a "Buckyball". In chemistry this structure is called a fullerene ("Buckminsterfullerene") and has formula C_60.

Tom Karzes, Polyhedron Coordination Sequences
N. J. A. Sloane, Illustration showing the 60 vertices and their distance from a base vertex.
Index entries for coordination sequences

Cf. A108997, A118785, A329499.

A330555 Coordination sequence for 1-skeleton of icosidodecahedron.

Original entry on oeis.org

1, 4, 8, 8, 8, 1

Offset: 0

N. J. A. Sloane, Dec 24 2019

The King-Canfield reference shows how this polyhedron is involved in the solution of the quintic equation using elliptic functions.

Tom Karzes, Polyhedron Coordination Sequences
R. B. King, and E. R. Canfield, Icosahedral symmetry and the quintic equation, Computers & Mathematics with Applications 24.3 (1992): 13-28. See Fig. 3.
N. J. A. Sloane, Vertices of icosidodecahedron labeled with distances from a base vertex (labeled 0), except for the antipode to the base vertex which is labeled A.

Cf. A329500.

A330564 Coordination sequence for 1-skeleton of small stellated dodecahedron with respect to a trivalent node (a valley).

Original entry on oeis.org

1, 6, 9, 9, 6, 1

Offset: 0

N. J. A. Sloane, Jan 01 2020

The coordination sequence with respect to a pentavalent node (a peak) is 1,5,10,10,5,1. Since this coordination sequence is a row of Pascal's triangle it does not have its own entry.

H. M. Cundy and A. P. Rollett, Mathematical Models, Oxford, 1st. ed., corrected 1957, pp. 84-85.

Tom Karzes, Polyhedron Coordination Sequences

Cf. A329500, A330555.

A329770 Coordination sequence for trivalent vertex in 1-skeleton of triakis icosahedron.

Original entry on oeis.org

1, 3, 15, 12, 1

Offset: 0

N. J. A. Sloane, Nov 25 2019

Tom Karzes, Polyhedron Coordination Sequences
N. J. A. Sloane, Triakis icosahedron showing distances from a trivalent vertex. The base vertex is labeled 0, and the vertices at distances 1, 2, and 4 are marked, as are some vertices at distance 3. Unlabeled vertices should also be labeled 3. [This image was constructed by adding labels to an image on the Wikipedia page.]
Wikipedia, Triakis icosahedron

See A329772 for a 10-valent vertex.

A329772 Coordination sequence for 10-valent vertex in 1-skeleton of triakis icosahedron.

Original entry on oeis.org

1, 10, 15, 6

Offset: 0

N. J. A. Sloane, Nov 25 2019

Tom Karzes, Polyhedron Coordination Sequences
N. J. A. Sloane, Triakis icosahedron showing distances from a 10-valent vertex. The base vertex is labeled 0, and the vertices at distances 1 and 3 are marked. The 15 unlabeled vertices should have the label 2. [This image was constructed by adding labels to an image on the Wikipedia page.]
Wikipedia, Triakis icosahedron

See A329770 for a trivalent vertex.

A331055 Coordination sequence for tetravalent vertex in 1-skeleton of disdyakis triacontahedron.

Original entry on oeis.org

1, 4, 16, 16, 16, 8, 1

Offset: 0

N. J. A. Sloane, Jan 08 2020

Computed by Tom Karzes (see link).

Tom Karzes, Polyhedron Coordination Sequences

Cf. A331056, A331057.

A331056 Coordination sequence for hexavalent vertex in 1-skeleton of disdyakis triacontahedron.

Original entry on oeis.org

1, 6, 18, 12, 18, 6, 1

Offset: 0

N. J. A. Sloane, Jan 08 2020

Computed by Tom Karzes (see link).

Tom Karzes, Polyhedron Coordination Sequences

Cf. A331055, A331057.

A331057 Coordination sequence for 10-valent vertex in 1-skeleton of disdyakis triacontahedron.

Original entry on oeis.org

1, 10, 10, 20, 10, 10, 1

Offset: 0

N. J. A. Sloane, Jan 08 2020

Computed by Tom Karzes (see link).

Tom Karzes, Polyhedron Coordination Sequences

Cf. A331055, A331056.

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A161435 Number of reduced words of length n in the Weyl group A_3 (or D_3).

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A161696 Number of reduced words of length n in the Weyl group B_3.

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A329500 Coordination sequence for 1-skeleton of truncated icosahedron or "Buckyball".

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A330555 Coordination sequence for 1-skeleton of icosidodecahedron.

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A330564 Coordination sequence for 1-skeleton of small stellated dodecahedron with respect to a trivalent node (a valley).

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A329770 Coordination sequence for trivalent vertex in 1-skeleton of triakis icosahedron.

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A329772 Coordination sequence for 10-valent vertex in 1-skeleton of triakis icosahedron.

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A331055 Coordination sequence for tetravalent vertex in 1-skeleton of disdyakis triacontahedron.

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A331056 Coordination sequence for hexavalent vertex in 1-skeleton of disdyakis triacontahedron.

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A331057 Coordination sequence for 10-valent vertex in 1-skeleton of disdyakis triacontahedron.

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