A168682 Number of reduced words of length n in Coxeter group on 5 generators S_i with relations (S_i)^2 = (S_i S_j)^17 = I.

1, 5, 20, 80, 320, 1280, 5120, 20480, 81920, 327680, 1310720, 5242880, 20971520, 83886080, 335544320, 1342177280, 5368709120, 21474836470, 85899345840, 343597383210, 1374389532240, 5497558126560, 21990232496640, 87960929948160

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

The initial terms coincide with those of A003947, although the two sequences are eventually different.

First disagreement at index 17: a(17) = 21474836470, A003947(17) = 21474836480. - Klaus Brockhaus, Mar 30 2011

Computed with MAGMA using commands similar to those used to compute A154638.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..500
• Index entries for linear recurrences with constant coefficients, signature (3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,-6).

Crossrefs

Cf. A003947 (G.f.: (1+x)/(1-4*x)).

Programs

Magma

R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( (1+t)*(1-t^17)/(1 -4*t +9*t^17 -6*t^18) )); // G. C. Greubel, Feb 22 2021

Mathematica

CoefficientList[Series[(1+t)*(1-t^17)/(1 -4*t +9*t^17 -6*t^18), {t, 0, 40}], t] (* G. C. Greubel, Aug 03 2016, Feb 22 2021 *)
coxG[{17, 6, -3, 40}] (* The coxG program is at A169452 *) (* G. C. Greubel, Feb 22 2021 *)

PARI

Vec(Pol(vector(18,i,if(i<2||i>17,1,2))) / Pol(vector(18,i,if(i<2,6,i>17,1,-3)))+O(x^99)) \\ Charles R Greathouse IV, Aug 03 2016

Sage

def A168682_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+t)*(1-t^17)/(1 -4*t +9*t^17 -6*t^18) ).list()
A168682_list(40) # G. C. Greubel, Feb 22 2021

Formula

G.f.: (t^17 + 2*t^16 + 2*t^15 + 2*t^14 + 2*t^13 + 2*t^12 + 2*t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1) / (6*t^17 - 3*t^16 - 3*t^15 - 3*t^14 - 3*t^13 - 3*t^12 - 3*t^11 - 3*t^10 - 3*t^9 - 3*t^8 - 3*t^7 - 3*t^6 - 3*t^5 - 3*t^4 - 3*t^3 - 3*t^2 - 3*t + 1).

G.f.: (1+t)*(1-t^17)/(1 -4*t +9*t^17 -6*t^18). - G. C. Greubel, Feb 22 2021