A164348 Number of reduced words of length n in Coxeter group on 48 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

1, 48, 2256, 106032, 4983504, 234224688, 11008559208, 517402229760, 24317902308096, 1142941291421184, 53718235195007232, 2524756795581284352, 118663557238871024856, 5577186619014877732560, 262127744246735162576688

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170767, 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..595
• Index entries for linear recurrences with constant coefficients, signature (46, 46, 46, 46, 46, -1081).

Programs

GAP

a:=[48, 2256, 106032, 4983504, 234224688, 11008559208];; for n in [7..20] do a[n]:=46*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -1081*a[n-6]; od; Concatenation([1], a); # G. C. Greubel, Aug 24 2019

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+t)*(1-t^6)/(1-47*t+1127*t^6-1081*t^7) )); // G. C. Greubel, Aug 24 2019

Maple

seq(coeff(series((1+t)*(1-t^6)/(1-47*t+1127*t^6-1081*t^7), t, n+1), t, n), n = 0..20); # G. C. Greubel, Aug 24 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^6)/(1-47*t+1127*t^6-1081*t^7), {t, 0, 20}], t] (* G. C. Greubel, Sep 15 2017 *)
coxG[{6, 1081, -46}] (* The coxG program is at A169452 *) (* G. C. Greubel, Aug 24 2019 *)

PARI

my(t='t+O('t^20)); Vec((1+t)*(1-t^6)/(1-47*t+1127*t^6-1081*t^7)) \\ G. C. Greubel, Sep 15 2017

Sage

def A164348_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^6)/(1-47*t+1127*t^6-1081*t^7)).list()
A164348_list(20) # G. C. Greubel, Aug 24 2019

Formula

G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(1081*t^6 - 46*t^5 - 46*t^4 - 46*t^3 - 46*t^2 - 46*t + 1).

a(n) = -1081*a(n-6) + 46*Sum_{k=1..5} a(n-k). - Wesley Ivan Hurt, May 07 2021