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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.

%I A166500 #15 Aug 03 2024 11:19:02
%S A166500 1,6,30,150,750,3750,18750,93750,468750,2343750,11718750,58593750,
%T A166500 292968735,1464843600,7324217640,36621086400,183105423000,
%U A166500 915527070000,4577635125000,22888174500000,114440866875000,572204306250000
%N 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.
%C A166500 The initial terms coincide with those of A003948, although the two sequences are eventually different.
%C A166500 Computed with MAGMA using commands similar to those used to compute A154638.
%H A166500 G. C. Greubel, <a href="/A166500/b166500.txt">Table of n, a(n) for n = 0..500</a>
%H A166500 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (4,4,4,4,4,4,4,4,4,4,4,-10).
%F A166500 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).
%F A166500 From _G. C. Greubel_, Aug 03 2024: (Start)
%F A166500 a(n) = 4*Sum_{j=1..11} a(n-j) - 10*a(n-12).
%F A166500 G.f.: (1+x)*(1-x^12)/(1 - 5*x + 14*x^12 - 10*x^13). (End)
%t A166500 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 *)
%t A166500 coxG[{12, 10, -4, 30}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Aug 03 2024 *)
%o A166500 (Magma)
%o A166500 R<x>:=PowerSeriesRing(Integers(), 30);
%o A166500 f:= func< p,q,x | (1+x)*(1-x^12)/(1-(q+1)*x+(p+q)*x^12-p*x^13) >;
%o A166500 Coefficients(R!( f(10,4,x) )); // _G. C. Greubel_, Aug 03 2024
%o A166500 (SageMath)
%o A166500 def f(p,q,x): return (1+x)*(1-x^12)/(1-(q+1)*x+(p+q)*x^12-p*x^13)
%o A166500 def A166500_list(prec):
%o A166500     P.<x> = PowerSeriesRing(ZZ, prec)
%o A166500     return P( f(10,4,x) ).list()
%o A166500 A166500_list(30) # _G. C. Greubel_, Aug 03 2024
%Y A166500 Cf. A003948, A154638, A169452.
%K A166500 nonn
%O A166500 0,2
%A A166500 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009