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A166551 Number of reduced words of length n in Coxeter group on 11 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

%I A166551 #15 Aug 24 2024 02:05:59
%S A166551 1,11,110,1100,11000,110000,1100000,11000000,110000000,1100000000,
%T A166551 11000000000,110000000000,1099999999945,10999999998900,
%U A166551 109999999983555,1099999999781100,10999999997266500,109999999967220000
%N A166551 Number of reduced words of length n in Coxeter group on 11 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.
%C A166551 The initial terms coincide with those of A003953, although the two sequences are eventually different.
%C A166551 Computed with MAGMA using commands similar to those used to compute A154638.
%H A166551 G. C. Greubel, <a href="/A166551/b166551.txt">Table of n, a(n) for n = 0..500</a>
%H A166551 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (9,9,9,9,9,9,9,9,9,9,9,-45).
%F A166551 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)/(45*t^12 - 9*t^11 - 9*t^10 - 9*t^9 - 9*t^8 - 9*t^7 - 9*t^6 - 9*t^5 - 9*t^4 - 9*t^3 - 9*t^2 - 9*t + 1).
%F A166551 From _G. C. Greubel_, Aug 23 2024: (Start)
%F A166551 a(n) = 9*Sum_{j=1..11} a(n-j) - 45*a(n-12).
%F A166551 G.f.: (1+x)*(1-x^12)/(1 - 10*x + 54*x^12 - 45*x^13). (End)
%t A166551 CoefficientList[Series[(1+t)*(1-t^12)/(1-10*t+54*t^12-45*t^13), {t, 0, 50}], t] (* _G. C. Greubel_, May 17 2016; Aug 23 2024 *)
%t A166551 coxG[{12,45,-9}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, Jul 20 2020 *)
%o A166551 (Magma)
%o A166551 R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^12)/(1-10*x+54*x^12-45*x^13) )); // _G. C. Greubel_, Aug 23 2024
%o A166551 (SageMath)
%o A166551 def A166551_list(prec):
%o A166551     P.<x> = PowerSeriesRing(ZZ, prec)
%o A166551     return P( (1+x)*(1-x^12)/(1-10*x+54*x^12-45*x^13) ).list()
%o A166551 A166551_list(30) # _G. C. Greubel_, Aug 23 2024
%Y A166551 Cf. A003953, A154638, A169452.
%K A166551 nonn
%O A166551 0,2
%A A166551 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009