A164365 Number of reduced words of length n in Coxeter group on 6 generators S_i with relations (S_i)^2 = (S_i S_j)^7 = I.

1, 6, 30, 150, 750, 3750, 18750, 93735, 468600, 2342640, 11711400, 58548000, 292695000, 1463250000, 7315125210, 36570003000, 182821904040, 913968987000, 4569142377000, 22842199635000, 114193439625000, 570879418872060

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A003948, 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 (4,4,4,4,4,4,-10).

Programs

GAP

a:=[6, 30, 150, 750, 3750, 18750, 93735];; for n in [8..30] do a[n]:=4*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]+a[n-6]) -10*a[n-7]; od; Concatenation([1], a); # G. C. Greubel, Aug 28 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^7)/(1-5*t+14*t^7-10*t^8) )); // G. C. Greubel, Aug 28 2019

Maple

seq(coeff(series((1+t)*(1-t^7)/(1-5*t+14*t^7-10*t^8), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 28 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^7)/(1-5*t+14*t^7-10*t^8), {t, 0, 30}], t] (* G. C. Greubel, Sep 15 2017 *)
coxG[{7, 10, -4}] (* The coxG program is at A169452 *) (* G. C. Greubel, Aug 28 2019 *)

PARI

my(t='t+O('t^30)); Vec((1+t)*(1-t^7)/(1-5*t+14*t^7-10*t^8)) \\ G. C. Greubel, Sep 15 2017

Sage

def A164365_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^7)/(1-5*t+14*t^7-10*t^8)).list()
A164365_list(30) # G. C. Greubel, Aug 28 2019

Formula

G.f.: (t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(10*t^7 - 4*t^6 - 4*t^5 - 4*t^4 - 4*t^3 - 4*t^2 - 4*t + 1).

a(n) = -10*a(n-7) + 4*Sum_{k=1..6} a(n-k). - Wesley Ivan Hurt, May 11 2021