A164050 Number of reduced words of length n in Coxeter group on 34 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

1, 34, 1122, 37026, 1221858, 40321314, 1330602801, 43909873920, 1449025228992, 47817812414592, 1577987144990784, 52073553849901056, 1718426553198820080, 56708052368589946368, 1871364939893753424384, 61755017003604231740928

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170753, 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..655
• Index entries for linear recurrences with constant coefficients, signature (32,32,32,32,32,-528).

Programs

GAP

a:=[34, 1122, 37026, 1221858, 40321314, 1330602801];; for n in [7..30] do a[n]:=32*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -528*a[n-6]; od; Concatenation([1], a); # G. C. Greubel, Aug 13 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-33*t+560*t^6-528*t^7) )); // G. C. Greubel, Aug 13 2019

Maple

seq(coeff(series((1+t)*(1-t^6)/(1-33*t+560*t^6-528*t^7), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 13 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^6)/(1-33*t+560*t^6-528*t^7), {t,0,30}], t] (* G. C. Greubel, Sep 08 2017 *)
LinearRecurrence[{32,32,32,32,32, -528}, {1,34,1122,37026, 1221858, 40321314, 1330602801}, 21] (* Vincenzo Librandi, Sep 09 2017 *)
coxG[{6, 528, -2}] (* The coxG program is at A169452 *) (* G. C. Greubel, Aug 13 2019 *)

PARI

my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-33*t+560*t^6-528*t^7)) \\ G. C. Greubel, Sep 08 2017

Sage

def A164050_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^6)/(1-33*t+560*t^6-528*t^7)).list()
A164050_list(30) # G. C. Greubel, Aug 13 2019

Formula

G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(528*t^6 - 32*t^5 - 32*t^4 - 32*t^3 - 32*t^2 - 32*t + 1).

a(n) = -528*a(n-6) + 32*Sum_{k=1..5} a(n-k). - Wesley Ivan Hurt, May 11 2021