A168685 Number of reduced words of length n in Coxeter group on 8 generators S_i with relations (S_i)^2 = (S_i S_j)^17 = I.

1, 8, 56, 392, 2744, 19208, 134456, 941192, 6588344, 46118408, 322828856, 2259801992, 15818613944, 110730297608, 775112083256, 5425784582792, 37980492079544, 265863444556780, 1861044111897264, 13027308783279504

Offset

0,2

Comments

The initial terms coincide with those of A003950, although the two sequences are eventually different.

First disagreement at index 17: a(17) = 265863444556780, A003950(17) = 265863444556808. - 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 (6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,-21).

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)/ (21*t^17 - 6*t^16 - 6*t^15 - 6*t^14 - 6*t^13 - 6*t^12 - 6*t^11 - 6*t^10 - 6*t^9 - 6*t^8 - 6*t^7 - 6*t^6 - 6*t^5 - 6*t^4 - 6*t^3 - 6*t^2 - 6*t + 1).

G.f.: (1+t)*(1-t^17)/(1 - 7*t + 27*t^17 - 21*t^18). - G. C. Greubel, Mar 24 2021

Mathematica

CoefficientList[Series[(1+t)*(1-t^17)/(1 -7*t +27*t^17 -21*t^18), {t, 0, 40}], t] (* G. C. Greubel, Aug 03 2016; Mar 24 2021 *)
coxG[{17, 21, -6, 40}] (* The coxG program is at A169452 *) (* G. C. Greubel, Mar 24 2021 *)

Prog

(Magma)
R<t>:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( (1+t)*(1-t^17)/(1 -7*t +27*t^17 -21*t^18) )); // G. C. Greubel, Mar 24 2021
(Sage)
def A168685_list(prec):
    P.<t> = PowerSeriesRing(ZZ, prec)
    return P( (1+t)*(1-t^17)/(1 -7*t +27*t^17 -21*t^18) ).list()
A168685_list(40) # G. C. Greubel, Mar 24 2021

Crossrefs

Cf. A003950 (g.f.: (1+x)/(1-7*x)).

Sequence in context: A167109 A167653 A167899 * A168733 A168781 A168829

Adjacent sequences: A168682 A168683 A168684 * A168686 A168687 A168688

Keyword

nonn

Author

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

Status

approved