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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.

%I A166495 #15 Aug 03 2024 01:51:02
%S A166495 1,5,20,80,320,1280,5120,20480,81920,327680,1310720,5242880,20971510,
%T A166495 83886000,335543850,1342174800,5368696800,21474777600,85899072000,
%U A166495 343596134400,1374383923200,5497533235200,21990123110400,87960453120000
%N 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.
%C A166495 The initial terms coincide with those of A003947, although the two sequences are eventually different.
%C A166495 Computed with MAGMA using commands similar to those used to compute A154638.
%H A166495 G. C. Greubel, <a href="/A166495/b166495.txt">Table of n, a(n) for n = 0..500</a>
%H A166495 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (3,3,3,3,3,3,3,3,3,3,3,-6).
%F A166495 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).
%F A166495 From _G. C. Greubel_, Aug 02 2024: (Start)
%F A166495 a(n) = 3*Sum_{j=1..11} a(n-j) - 6*a(n-12).
%F A166495 G.f.: (1+x)*(1-x^12)/(1 - 4*x + 9*x^12 - 6*x^13). (End)
%t A166495 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 *)
%t A166495 coxG[{12,6,-3,30}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, Feb 19 2018 *)
%o A166495 (Magma)
%o A166495 R<x>:=PowerSeriesRing(Integers(), 30);
%o A166495 f:= func< p,q,x | (1+x)*(1-x^12)/(1-(q+1)*x+(p+q)*x^12-p*x^13) >;
%o A166495 Coefficients(R!( f(6,3,x) )); // _G. C. Greubel_, Aug 02 2024
%o A166495 (SageMath)
%o A166495 def f(p,q,x): return (1+x)*(1-x^12)/(1-(q+1)*x+(p+q)*x^12-p*x^13)
%o A166495 def A166495_list(prec):
%o A166495     P.<x> = PowerSeriesRing(ZZ, prec)
%o A166495     return P( f(6,3,x) ).list()
%o A166495 A166495_list(30) # _G. C. Greubel_, Aug 02 2024
%Y A166495 Cf. A003947, A154638, A169452.
%K A166495 nonn
%O A166495 0,2
%A A166495 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009