A166500 - OEIS

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

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OFFSET

0,2

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).

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)

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 *)

PROG

(Magma)

R<x>:=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

(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.<x> = PowerSeriesRing(ZZ, prec)

return P( f(10, 4, x) ).list()

A166500_list(30) # G. C. Greubel, Aug 03 2024