A163953 Number of reduced words of length n in Coxeter group on 9 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

1, 9, 72, 576, 4608, 36864, 294876, 2358720, 18867492, 150921792, 1207229184, 9656672256, 77244089580, 617878417968, 4942433025684, 39534710232528, 316239654648960, 2529613056079872, 20234471292326844, 161856307428494112

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,-28).

Programs

GAP

a:=[9,72,576,4608,36864,294876];; for n in [7..30] do a[n]:=7*(a[n-1]+a[n-2]+a[n-3]+a[n-4]+a[n-5]) -28*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)/(1-8*t+35*t^6-28*t^7) )); // G. C. Greubel, Aug 10 2019

Maple

seq(coeff(series((1+t)*(1-t^6)/(1-8*t+35*t^6-28*t^7), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 10 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^6)/(1-8*t+35*t^6-28*t^7), {t, 0, 30}], t] (* G. C. Greubel, Aug 13 2017 *)
coxG[{6, 28, -7}] (* The coxG program is at A169452 *) (* G. C. Greubel, Aug 10 2019 *)

PARI

my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-8*t+35*t^6-28*t^7)) \\ G. C. Greubel, Aug 13 2017

Sage

def A163953_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^6)/(1-8*t+35*t^6-28*t^7)).list()
A163953_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)/(28*t^6 - 7*t^5 - 7*t^4 - 7*t^3 - 7*t^2 - 7*t + 1).

a(n) = -28*a(n-6) + 7*Sum_{k=1..5} a(n-k). - Wesley Ivan Hurt, May 11 2021