A166541 Number of reduced words of length n in Coxeter group on 9 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

1, 9, 72, 576, 4608, 36864, 294912, 2359296, 18874368, 150994944, 1207959552, 9663676416, 77309411292, 618475290048, 4947802318116, 39582418526784, 316659348069120, 2533274783391744, 20266198257844224, 162129585988435968

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.

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,-28).

Crossrefs

Cf. A003951, A154638, A169452.

Programs

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^12)/(1-8*x+35*x^12-28*x^13) )); // G. C. Greubel, Aug 23 2024

Mathematica

CoefficientList[Series[(1+t)*(1-t^12)/(1-8*t+35*t^12-28*t^13), {t, 0, 50}], t] (* G. C. Greubel, May 16 2016; Aug 23 2024 *)
coxG[{12,28,-7, 30}] (* The coxG program is at A169452 *) (* G. C. Greubel, Aug 23 2024 *)

SageMath

def A166541_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^12)/(1-8*x+35*x^12-28*x^13) ).list()
A166541_list(30) # G. C. Greubel, Aug 23 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)/(28*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).

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

a(n) = 7*Sum_{j=1..11} a(n-j) - 28*a(n-13).

G.f.: (1+x)*(1-x^12)/(1 - 8*x + 35*x^12 - 28*x^13). (End)