A168688 Number of reduced words of length n in Coxeter group on 11 generators S_i with relations (S_i)^2 = (S_i S_j)^17 = I.

1, 11, 110, 1100, 11000, 110000, 1100000, 11000000, 110000000, 1100000000, 11000000000, 110000000000, 1100000000000, 11000000000000, 110000000000000, 1100000000000000, 11000000000000000, 109999999999999945

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A003953, although the two sequences are eventually different.

First disagreement at index 17: a(17) = 109999999999999945, A003953(17) = 110000000000000000. - 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 (9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,-45).

Crossrefs

Cf. A003953 (g.f.: (1+x)/(1-10*x)).

Programs

Magma

R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( (1+t)*(1-t^17)/(1 -10*t +54*t^17 -45*t^18) )); // G. C. Greubel, Mar 24 2021

Mathematica

CoefficientList[Series[(1+t)*(1-t^17)/(1 -10*t +54*t^17 -45*t^18), {t, 0, 50}], t] (* G. C. Greubel, Aug 03 2016; Mar 24 2021 *)
coxG[{17, 45, -9, 40}] (* The coxG program is at A169452 *) (* G. C. Greubel, Mar 24 2021 *)

Sage

def A168688_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+t)*(1-t^17)/(1 -10*t +54*t^17 -45*t^18) ).list()
A168688_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)/ (45*t^17 - 9*t^16 - 9*t^15 - 9*t^14 - 9*t^13 - 9*t^12 - 9*t^11 - 9*t^10 - 9*t^9 - 9*t^8 - 9*t^7 - 9*t^6 - 9*t^5 - 9*t^4 - 9*t^3 - 9*t^2 - 9*t + 1).

G.f.: (1+t)*(1-t^17)/(1 - 10*t + 54*t^17 -4 5*t^18). - G. C. Greubel, Mar 24 2021