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Number of reduced words of length n in Coxeter group on 9 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.
1

%I #13 Aug 23 2024 22:08:02

%S 1,9,72,576,4608,36864,294912,2359296,18874368,150994944,1207959552,

%T 9663676416,77309411292,618475290048,4947802318116,39582418526784,

%U 316659348069120,2533274783391744,20266198257844224,162129585988435968

%N Number of reduced words of length n in Coxeter group on 9 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

%C The initial terms coincide with those of A003951, although the two sequences are eventually different.

%C Computed with MAGMA using commands similar to those used to compute A154638.

%H G. C. Greubel, <a href="/A166541/b166541.txt">Table of n, a(n) for n = 0..500</a>

%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (7,7,7,7,7,7,7,7,7,7, 7,-28).

%F 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)/(28*t^12 - 7*t^11 - 7*t^10 - 7*t^9 - 7*t^8 - 7*t^7 - 7*t^6 - 7*t^5 - 7*t^4 - 7*t^3 - 7*t^2 - 7*t + 1).

%F From _G. C. Greubel_, Aug 23 2024: (Start)

%F a(n) = 7*Sum_{j=1..11} a(n-j) - 28*a(n-13).

%F G.f.: (1+x)*(1-x^12)/(1 - 8*x + 35*x^12 - 28*x^13). (End)

%t CoefficientList[Series[(1+t)*(1-t^12)/(1-8*t+35*t^12-28*t^13), {t, 0, 50}], t] (* _G. C. Greubel_, May 16 2016; Aug 23 2024 *)

%t coxG[{12,28,-7, 30}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Aug 23 2024 *)

%o (Magma)

%o R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^12)/(1-8*x+35*x^12-28*x^13) )); // _G. C. Greubel_, Aug 23 2024

%o (SageMath)

%o def A166541_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P( (1+x)*(1-x^12)/(1-8*x+35*x^12-28*x^13) ).list()

%o A166541_list(30) # _G. C. Greubel_, Aug 23 2024

%Y Cf. A003951, A154638, A169452.

%K nonn

%O 0,2

%A _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009