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A166568 Number of reduced words of length n in Coxeter group on 14 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

%I A166568 #16 Dec 03 2024 08:22:56
%S A166568 1,14,182,2366,30758,399854,5198102,67575326,878479238,11420230094,
%T A166568 148462991222,1930018885886,25090245516427,326173191712368,
%U A166568 4240251492245496,55123269398992704,716602502184321480
%N A166568 Number of reduced words of length n in Coxeter group on 14 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.
%C A166568 The initial terms coincide with those of A170733, although the two sequences are eventually different.
%C A166568 Computed with MAGMA using commands similar to those used to compute A154638.
%H A166568 G. C. Greubel, <a href="/A166568/b166568.txt">Table of n, a(n) for n = 0..500</a>
%H A166568 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (12,12,12,12,12,12,12,12,12,12,12,-78).
%F A166568 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)/(78*t^12 - 12*t^11 - 12*t^10 - 12*t^9 - 12*t^8 - 12*t^7 - 12*t^6 - 12*t^5 - 12*t^4 - 12*t^3 - 12*t^2 - 12*t +1).
%F A166568 From _G. C. Greubel_, Dec 03 2024: (Start)
%F A166568 a(n) = 12*Sum_{j=1..11} a(n-j) - 78*a(n-12).
%F A166568 G.f.: (1+x)*(1-x^12)/(1 - 13*x + 90*x^12 - 78*x^13). (End)
%t A166568 coxG[{12,78,-12}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, Feb 03 2015 *)
%t A166568 CoefficientList[Series[(1+t)*(1-t^12)/(1-13*t+90*t^12-78*t^13), {t,0,50}], t] (* _G. C. Greubel_, May 17 2016; Dec 03 2024 *)
%o A166568 (Magma)
%o A166568 R<x>:=PowerSeriesRing(Integers(), 40);
%o A166568 Coefficients(R!( (1+x)*(1-x^12)/(1-13*x+90*x^12-78*x^13) )); // _G. C. Greubel_, Dec 03 2024
%o A166568 (SageMath)
%o A166568 def A166568_list(prec):
%o A166568     P.<x> = PowerSeriesRing(ZZ, prec)
%o A166568     return P( (1+x)*(1-x^12)/(1-13*x+90*x^12-78*x^13) ).list()
%o A166568 A166568_list(40) # _G. C. Greubel_, Dec 03 2024
%Y A166568 Cf. A154638, A169452, A170733.
%K A166568 nonn
%O A166568 0,2
%A A166568 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009