A164373 Number of reduced words of length n in Coxeter group on 8 generators S_i with relations (S_i)^2 = (S_i S_j)^7 = I.

1, 8, 56, 392, 2744, 19208, 134456, 941164, 6587952, 46114320, 322790832, 2259469968, 15815828784, 110707574544, 774930433956, 5424354927432, 37969377752376, 265777897314888, 1860391054122552, 13022357800350024

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, 6, -21).

Programs

GAP

a:=[8, 56, 392, 2744, 19208, 134456, 941164];; for n in [8..30] do a[n]:=6*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]+a[n-6]) -21*a[n-7]; od; Concatenation([1], a); # G. C. Greubel, Aug 28 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^7)/(1-7*t+27*t^7-21*t^8) )); // G. C. Greubel, Aug 28 2019

Maple

seq(coeff(series((1+t)*(1-t^7)/(1-7*t+27*t^7-21*t^8), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 28 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^7)/(1-7*t+27*t^7-21*t^8), {t, 0, 30}], t] (* G. C. Greubel, Sep 17 2017 *)
coxG[{7, 21, -6}] (* The coxG program is at A169452 *) (* G. C. Greubel, Aug 28 2019 *)

PARI

my(t='t+O('t^30)); Vec((1+t)*(1-t^7)/(1-7*t+27*t^7-21*t^8)) \\ G. C. Greubel, Sep 17 2017

Sage

def A164373_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^7)/(1-7*t+27*t^7-21*t^8)).list()
A164373_list(30) # G. C. Greubel, Aug 28 2019

Formula

G.f.: (t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(21*t^7 - 6*t^6 - 6*t^5 - 6*t^4 - 6*t^3 - 6*t^2 - 6*t + 1).

a(n) = -21*a(n-7) + 6*Sum_{k=1..6} a(n-k). - Wesley Ivan Hurt, May 11 2021