A164354 Number of reduced words of length n in Coxeter group on 5 generators S_i with relations (S_i)^2 = (S_i S_j)^7 = I.

1, 5, 20, 80, 320, 1280, 5120, 20470, 81840, 327210, 1308240, 5230560, 20912640, 83612160, 334295130, 1336566780, 5343813270, 21365442180, 85422543120, 341533342080, 1365506334720, 5459518355670, 21828050092440, 87272125451010

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 (3,3,3,3,3,3,-6).

Programs

GAP

a:=[5, 20, 80, 320, 1280, 5120, 20470];; for n in [8..30] do a[n]:=3*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]+a[n-6]) -6*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)/(1-4*t+9*t^7-6*t^8) )); // G. C. Greubel, Aug 28 2019

Maple

seq(coeff(series((1+t)*(1-t^7)/(1-4*t+9*t^7-6*t^8), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 28 2019

Mathematica

coxG[{7,6,-3,30}] (* The coxG program is at A169452 *) (* Harvey P. Dale, May 01 2017 *)
CoefficientList[Series[(1+t)*(1-t^7)/(1-4*t+9*t^7-6*t^8), {t, 0, 30}], t] (* G. C. Greubel, Sep 15 2017 *)

PARI

my(t='t+O('t^30)); Vec((1+t)*(1-t^7)/(1-4*t+9*t^7-6*t^8)) \\ G. C. Greubel, Sep 15 2017

Sage

def A164354_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^7)/(1-4*t+9*t^7-6*t^8)).list()
A164354_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)/(6*t^7 - 3*t^6 - 3*t^5 - 3*t^4 - 3*t^3 - 3*t^2 - 3*t + 1).

a(n) = -6*a(n-7) + 3*Sum_{k=1..6} a(n-k). - Wesley Ivan Hurt, May 11 2021