A166538 Number of reduced words of length n in Coxeter group on 8 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

1, 8, 56, 392, 2744, 19208, 134456, 941192, 6588344, 46118408, 322828856, 2259801992, 15818613916, 110730297216, 775112079168, 5425784544768, 37980491747520, 265863441771648, 1861044089174592, 13027308601633536

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A003950, 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 (6,6,6,6,6,6,6,6,6,6,6,-21).

Crossrefs

Cf. A003950, A154638, A169452.

Programs

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^12)/(1-7*x+27*x^12-21*x^13) )); // G. C. Greubel, Aug 23 2024

Mathematica

CoefficientList[Series[(1+t)*(1-t^12)/(1-7*t+27*t^12-21*t^13), {t, 0, 50}], t] (* G. C. Greubel, May 16 2016; Aug 23 2024 *)
coxG[{12,21,-6}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Oct 24 2016 *)

SageMath

def A166538_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^12)/(1-7*x+27*x^12-21*x^13) ).list()
A166538_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)/(21*t^12 - 6*t^11 - 6*t^10 - 6*t^9 - 6*t^8 - 6*t^7 - 6*t^6 - 6*t^5 - 6*t^4 - 6*t^3 - 6*t^2 - 6*t + 1).

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

a(n) = 6*Sum_{j=1..11} a(n-j) - 21*a(n-12).

G.f.: (1 + x)*(1 - x^12)/(1 - 7*x + 27*x^12 - 21*x^13). (End)