A166500 Number of reduced words of length n in Coxeter group on 6 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

1, 6, 30, 150, 750, 3750, 18750, 93750, 468750, 2343750, 11718750, 58593750, 292968735, 1464843600, 7324217640, 36621086400, 183105423000, 915527070000, 4577635125000, 22888174500000, 114440866875000, 572204306250000

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A003948, 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 (4,4,4,4,4,4,4,4,4,4,4,-10).

Crossrefs

Cf. A003948, A154638, A169452.

Programs

Magma

R:=PowerSeriesRing(Integers(), 30);
f:= func< p,q,x | (1+x)*(1-x^12)/(1-(q+1)*x+(p+q)*x^12-p*x^13) >;
Coefficients(R!( f(10,4,x) )); // G. C. Greubel, Aug 03 2024

Mathematica

With[{p=10, q=4}, CoefficientList[Series[(1+t)*(1-t^12)/(1 - (q+1)*t + (p+q)*t^12 - p*t^13), {t,0,40}], t]] (* G. C. Greubel, May 15 2016; Aug 02 2024 *)
coxG[{12, 10, -4, 30}] (* The coxG program is at A169452 *) (* G. C. Greubel, Aug 03 2024 *)

SageMath

def f(p,q,x): return (1+x)*(1-x^12)/(1-(q+1)*x+(p+q)*x^12-p*x^13)
def A166500_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( f(10,4,x) ).list()
A166500_list(30) # G. C. Greubel, Aug 03 2024

Formula

G.f.: (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^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).

From G. C. Greubel, Aug 03 2024: (Start)

a(n) = 4*Sum_{j=1..11} a(n-j) - 10*a(n-12).

G.f.: (1+x)*(1-x^12)/(1 - 5*x + 14*x^12 - 10*x^13). (End)