A164618 Number of reduced words of length n in Coxeter group on 14 generators S_i with relations (S_i)^2 = (S_i S_j)^7 = I.

1, 14, 182, 2366, 30758, 399854, 5198102, 67575235, 878476872, 11420184048, 148462193880, 1930005936768, 25090043590248, 326170130032656, 4240206014105334, 55122604391319192, 716592897791781192

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170733, 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..895
• Index entries for linear recurrences with constant coefficients, signature (12,12,12,12,12,12,-78).

Programs

GAP

a:=[14,182,2366,30758,399854,5198102,67575235];; for n in [8..20] do a[n]:=12*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]+a[n-6]) -78*a[n-6]; od; Concatenation([1], a); # G. C. Greubel, Sep 15 2019

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+t)*(1-t^7)/(1-13*t+90*t^7-78*t^8) )); // G. C. Greubel, Sep 15 2019

Maple

seq(coeff(series((1+t)*(1-t^7)/(1-13*t+90*t^7-78*t^8), t, n+1), t, n), n = 0..20); # G. C. Greubel, Sep 15 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^7)/(1-13*t+90*t^7-78*t^8), {t,0,20}],t] (* G. C. Greubel, Aug 10 2017 *)
coxG[{7, 78, -12}] (* The coxG program is at A169452 *) (* G. C. Greubel, Sep 15 2019 *)

PARI

my(t='t+O('t^20)); Vec((1+t)*(1-t^7)/(1-13*t+90*t^7-78*t^8)) \\ G. C. Greubel, Aug 10 2017

Sage

def A164618_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^7)/(1-13*t+90*t^7-78*t^8)).list()
A164618_list(20) # G. C. Greubel, Sep 15 2019

Formula

G.f.: (t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(78*t^7 - 12*t^6 - 12*t^5 - 12*t^4 - 12*t^3 - 12*t^2 - 12*t + 1).

a(n) = -78*a(n-7) + 12*Sum_{k=1..6} a(n-k). - Wesley Ivan Hurt, May 11 2021