A164375 Number of reduced words of length n in Coxeter group on 9 generators S_i with relations (S_i)^2 = (S_i S_j)^7 = I.

1, 9, 72, 576, 4608, 36864, 294912, 2359260, 18873792, 150988068, 1207886400, 9662946048, 77302407168, 618409967616, 4947205424364, 39577048871472, 316611634855572, 2532855030486480, 20262535861599360, 162097851871033344

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A003951, 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 (7,7,7,7,7,7,-28).

Programs

GAP

a:=[9, 72, 576, 4608, 36864, 294912, 2359260];; for n in [8..30] do a[n]:=7*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]+a[n-6]) -28*a[n-7]; od; Concatenation([1], a); # G. C. Greubel, Aug 10 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^7)/(1-8*t+35*t^7-28*t^8) )); // G. C. Greubel, Aug 10 2019

Maple

seq(coeff(series((1+t)*(1-t^7)/(1-8*t+35*t^7-28*t^8), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 10 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^7)/(1-8*t+35*t^7-28*t^8), {t, 0, 30}], t] (* G. C. Greubel, Sep 17 2017 *)
coxG[{7,28,-7}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Aug 20 2019 *)

PARI

my(t='t+O('t^30)); Vec((1+t)*(1-t^7)/(1-8*t+35*t^7-28*t^8)) \\ G. C. Greubel, Sep 17 2017

Sage

def A164375_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^7)/(1-8*t+35*t^7-28*t^8)).list()
A164375_list(30) # G. C. Greubel, Aug 10 2019

Formula

G.f.: (t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(28*t^7 - 7*t^6 - 7*t^5 - 7*t^4 - 7*t^3 - 7*t^2 - 7*t + 1).

a(n) = -28*a(n-7) + 7*Sum_{k=1..6} a(n-k). - Wesley Ivan Hurt, May 11 2021