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A161435 Number of reduced words of length n in the Weyl group A_3 (or D_3). 50
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
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.
REFERENCES
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.
LINKS
FORMULA
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.
MAPLE
# 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;
MATHEMATICA
CoefficientList[Series[(1 - x^2) (1 - x^3) (1 - x^4) / (1 - x)^3, {x, 0, 20}], x] (* Vincenzo Librandi, Aug 23 2016 *)
CROSSREFS
Sequence in context: A106117 A081498 A110279 * A354213 A343460 A224831
KEYWORD
nonn,easy
AUTHOR
John Cannon and N. J. A. Sloane, Nov 30 2009
STATUS
approved

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Last modified March 19 02:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)