A168686 Number of reduced words of length n in Coxeter group on 9 generators S_i with relations (S_i)^2 = (S_i S_j)^17 = I.

1, 9, 72, 576, 4608, 36864, 294912, 2359296, 18874368, 150994944, 1207959552, 9663676416, 77309411328, 618475290624, 4947802324992, 39582418599936, 316659348799488, 2533274790395868, 20266198323166656, 162129586585330980

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A003951, although the two sequences are eventually different.

First disagreement at index 17: a(17) = 2533274790395868, A003951(17) = 2533274790395904. - 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 (7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,-28).

Crossrefs

Cf. A003951 (g.f.: (1+x)/(1-8*x)).

Programs

Magma

R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( (1+t)*(1-t^17)/(1 -8*t +35*t^17 -28*t^18) )); // G. C. Greubel, Mar 24 2021

Mathematica

CoefficientList[Series[(1+t)*(1-t^17)/(1 -8*t +35*t^17 -28*t^18), {t, 0, 40}], t] (* G. C. Greubel, Aug 03 2016; Mar 24 2021 *)
coxG[{17, 28, -7, 40}] (* The coxG program is at A169452 *) (* G. C. Greubel, Mar 24 2021 *)

Sage

def A168686_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+t)*(1-t^17)/(1 -8*t +35*t^17 -28*t^18) ).list()
A168686_list(40) # G. C. Greubel, Mar 24 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)/ (28*t^17 - 7*t^16 - 7*t^15 - 7*t^14 - 7*t^13 - 7*t^12 - 7*t^11 - 7*t^10 - 7*t^9 - 7*t^8 - 7*t^7 - 7*t^6 - 7*t^5 - 7*t^4 - 7*t^3 - 7*t^2 - 7*t + 1).

G.f.: (1+t)*(1-t^17)/(1 - 8*t + 35*t^17 - 28*t^18). - G. C. Greubel, Mar 24 2021