A166369 Number of reduced words of length n in Coxeter group on 11 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

1, 11, 110, 1100, 11000, 110000, 1100000, 11000000, 110000000, 1100000000, 11000000000, 109999999945, 1099999998900, 10999999983555, 109999999781100, 1099999997266500, 10999999967220000, 109999999617750000

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.

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,-45).

Crossrefs

Cf. A003953, A154638, A169452.

Programs

Magma

R:=PowerSeriesRing(Integers(), 30);
Coefficients(R!( (1+x)*(1-x^11)/(1-10*x+54*x^11-45*x^12) )); // G. C. Greubel, Jul 23 2024

Mathematica

With[{p=45, q=9}, CoefficientList[Series[(1+t)*(1-t^11)/(1-(q+1)*t + (p+q)*t^11-p*t^12), {t,0,40}], t]] (* G. C. Greubel, May 10 2016; Jul 23 2024 *)
coxG[{11,45,-9}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jun 12 2016 *)

SageMath

def A166369_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^11)/(1-10*x+54*x^11-45*x^12) ).list()
A166369_list(30) # G. C. Greubel, Jul 23 2024

Formula

G.f.: (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^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).

From G. C. Greubel, Jul 23 2024: (Start)

a(n) = 9*Sum_{j=1..10} a(n-j) - 45*a(n-11).

G.f.: (1+t)*(1-t^11)/(1 - 10*t + 54*t^11 - 45*t^12). (End)