A168683 Number of reduced words of length n in Coxeter group on 6 generators S_i with relations (S_i)^2 = (S_i S_j)^17 = I.

1, 6, 30, 150, 750, 3750, 18750, 93750, 468750, 2343750, 11718750, 58593750, 292968750, 1464843750, 7324218750, 36621093750, 183105468750, 915527343735, 4577636718600, 22888183592640, 114440917961400, 572204589798000

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

The initial terms coincide with those of A003948, although the two sequences are eventually different.

First disagreement at index 17: a(17) = 915527343735, A003948(17) = 915527343750. - 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 (4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,-10).

Crossrefs

Cf. A003948 (G.f.: (1+x)/(1-5*x)).

Programs

Magma

R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( (1+t)*(1-t^17)/(1 -5*t +14*t^17 -10*t^18) )); // G. C. Greubel, Feb 22 2021

Mathematica

CoefficientList[Series[(1+t)*(1-t^17)/(1 -5*t +14*t^17 -10*t^18), {t, 0, 40}], t] (* G. C. Greubel, Aug 03 2016, Feb 22 2021 *)
coxG[{17,10,-4,30}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jun 09 2017 *)

Sage

def A168683_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+t)*(1-t^17)/(1 -5*t +14*t^17 -10*t^18) ).list()
A168683_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) / (10*t^17 - 4*t^16 - 4*t^15 - 4*t^14 - 4*t^13 - 4*t^12 - 4*t^11 - 4*t^10 - 4*t^9 - 4*t^8 - 4*t^7 - 4*t^6 - 4*t^5 - 4*t^4 - 4*t^3 - 4*t^2 - 4*t + 1).

G.f.: (1+t)*(1-t^17)/(1 -5*t +14*t^17 -10*t^18). - G. C. Greubel, Feb 22 2021