A166495 Number of reduced words of length n in Coxeter group on 5 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

1, 5, 20, 80, 320, 1280, 5120, 20480, 81920, 327680, 1310720, 5242880, 20971510, 83886000, 335543850, 1342174800, 5368696800, 21474777600, 85899072000, 343596134400, 1374383923200, 5497533235200, 21990123110400, 87960453120000

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A003947, 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 (3,3,3,3,3,3,3,3,3,3,3,-6).

Crossrefs

Cf. A003947, 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(6,3,x) )); // G. C. Greubel, Aug 02 2024

Mathematica

CoefficientList[Series[(1+t)*(1-t^12)/(1-4*t+9*t^12-6*t^13), {t, 0, 50}], t] (* G. C. Greubel, May 15 2016; Aug 02 2024 *)
coxG[{12,6,-3,30}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Feb 19 2018 *)

SageMath

def f(p,q,x): return (1+x)*(1-x^12)/(1-(q+1)*x+(p+q)*x^12-p*x^13)
def A166495_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( f(6,3,x) ).list()
A166495_list(30) # G. C. Greubel, Aug 02 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)/(6*t^12 - 3*t^11 - 3*t^10 - 3*t^9 - 3*t^8 - 3*t^7 - 3*t^6 - 3*t^5 - 3*t^4 - 3*t^3 - 3*t^2 - 3*t + 1).

From G. C. Greubel, Aug 02 2024: (Start)

a(n) = 3*Sum_{j=1..11} a(n-j) - 6*a(n-12).

G.f.: (1+x)*(1-x^12)/(1 - 4*x + 9*x^12 - 6*x^13). (End)