A163316 Number of reduced words of length n in Coxeter group on 5 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

1, 5, 20, 80, 320, 1270, 5040, 20010, 79440, 315360, 1251930, 4969980, 19730070, 78325380, 310939920, 1234384470, 4900319640, 19453527810, 77227563240, 306581745960, 1217083163130, 4831636082580, 19180864497870, 76145131089180

Offset: 0

Author

John Cannon and N. J. A. Sloane, Dec 03 2009

Keywords

• nonn

Comments

The initial terms coincide with those of A003947, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3, 3, 3, 3, -6).

Programs

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^5)/(1-4*x+9*x^5-6*x^6) )); // G. C. Greubel, May 12 2019

Mathematica

CoefficientList[Series[(1+x)*(1-x^5)/(1-4*x+9*x^5-6*x^6), {x, 0, 30}], x] (* or *) LinearRecurrence[{3,3,3,3,-6}, {1,5,20,80,320,1270}, 30] (* G. C. Greubel, Dec 18 2016 *)
coxG[{5, 6, -3}] (* The coxG program is at A169452 *) (* G. C. Greubel, May 12 2019 *)

PARI

my(x='x+O('x^30)); Vec((1+x)*(1-x^5)/(1-4*x+9*x^5-6*x^6)) \\ G. C. Greubel, Dec 18 2016

Sage

((1+x)*(1-x^5)/(1-4*x+9*x^5-6*x^6)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 12 2019

Formula

G.f.: (t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(6*t^5 - 3*t^4 - 3*t^3 - 3*t^2 - 3*t + 1).

a(n) = 3*a(n-1)+3*a(n-2)+3*a(n-3)+3*a(n-4)-6*a(n-5). - Wesley Ivan Hurt, May 10 2021