A163924 Number of reduced words of length n in Coxeter group on 8 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

1, 8, 56, 392, 2744, 19208, 134428, 940800, 6584256, 46080384, 322496832, 2257016832, 15795891636, 110548662840, 773682621768, 5414672451384, 37894967433288, 265210605012024, 1856095143363468, 12990012903371952

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A003950, 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 (6,6,6,6,6,-21).

Programs

GAP

a:=[8,56,392,2744,19208,134428];; for n in [7..30] do a[n]:=6*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -21*a[n-6]; od; Concatenation([1], a); # G. C. Greubel, Aug 10 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-7*t+27*t^6-21*t^7) )); // G. C. Greubel, Aug 10 2019

Maple

seq(coeff(series((1+t)*(1-t^6)/(1-7*t+27*t^6-21*t^7), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 10 2019

Mathematica

coxG[{6,21,-6}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jan 24 2016 *)
CoefficientList[Series[(1+t)*(1-t^6)/(1-7*t+27*t^6-21*t^7), {t,0,30}], t] (* G. C. Greubel, Aug 08 2017 *)

PARI

my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-7*t+27*t^6-21*t^7)) \\ G. C. Greubel, Aug 08 2017

Sage

def A163924_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^6)/(1-7*t+27*t^6-21*t^7)).list()
A163924_list(30) # G. C. Greubel, Aug 10 2019

Formula

G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(21*t^6 - 6*t^5 - 6*t^4 - 6*t^3 - 6*t^2 - 6*t + 1).

a(n) = -21*a(n-6) + 6*Sum_{k=1..5} a(n-k). - Wesley Ivan Hurt, May 11 2021