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Number of reduced words of length n in Coxeter group on 4 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I.
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%I #30 Sep 08 2022 08:45:46

%S 1,4,12,30,72,168,390,900,2076,4782,11016,25368,58422,134532,309804,

%T 713406,1642824,3783048,8711526,20060676,46195260,106377294,244963080,

%U 564094968,1298984214,2991269124,6888221772,15862029150,36526694472,84112781928,193692865350

%N Number of reduced words of length n in Coxeter group on 4 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I.

%C The initial terms coincide with those of A003946, although the two sequences are eventually different.

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

%C From _Bruno Berselli_, Dec 28 2015: (Start)

%C Also, expansion of b(2)*b(3)/(1 - 2*x - 2*x^2 + 3*x^3), where b(k) = (1-x^k)/(1-x).

%C This is also the Poincaré series [or Poincare series] for the quasi-Lannér diagram QL4_22 - see Table 7.8 in Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2009), or equivalently Table 6 in the shorter version, Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2010).

%C (End)

%H Vincenzo Librandi, <a href="/A162740/b162740.txt">Table of n, a(n) for n = 0..1000</a>

%H Maxim Chapovalov, Dimitry Leites, and Rafael Stekolshchik, <a href="http://arxiv.org/abs/0906.1596">The Poincaré series [or Poincare series] of the hyperbolic Coxeter groups with finite volume of fundamental domains</a>, arXiv:0906.1596 [math.RT], 2009, page 31.

%H Maxim Chapovalov, Dimitry Leites, and Rafael Stekolshchik, <a href="http://dx.doi.org/10.1142/S1402925110000842">The Poincaré series [or Poincare series] of the hyperbolic Coxeter groups with finite volume of fundamental domains</a>, Journal of Nonlinear Mathematical Physics 17.supp01 (2010), page 186.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-3).

%F G.f.: (x^3 + 2*x^2 + 2*x + 1)/(3*x^3 - 2*x^2 - 2*x + 1).

%F From _Bruno Berselli_, Dec 28 2015: (Start)

%F a(n) = 2*a(n-1) + 2*a(n-2) - 3*a(n-3) for n>3.

%F a(n) = -2 + ((-7+2*sqrt(13))*(1-sqrt(13))^n + (7+2*sqrt(13))*(1+sqrt(13))^n)/(3*sqrt(13)*2^(n-1)) for n>0. (End)

%F G.f.: (1+x)*(1-x^3)/(1 -3*x +5*x^3 -3*x^4). - _G. C. Greubel_, Apr 25 2019

%t CoefficientList[Series[(x^3+2x^2+2x+1)/(3x^3-2x^2-2x+1), {x, 0, 40}], x ] (* _Vincenzo Librandi_, Apr 29 2014 *)

%t coxG[{3, 3, -2, 40}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Apr 25 2019 *)

%o (Magma) m:=40; R<x>:=PowerSeriesRing(Integers(), m); b:=func<k|(1-x^k)/(1-x)>; Coefficients(R!(b(2)*b(3)/(1-2*x-2*x^2+3*x^3))); // _Bruno Berselli_, Dec 28 2015 - see Chapovalov et al.

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^3)/(1-3*x+5*x^3-3*x^4) )); // _G. C. Greubel_, Apr 25 2019

%o (PARI) my(x='x+O('x^40)); Vec((1+x)*(1-x^3)/(1-3*x+5*x^3-3*x^4)) \\ _G. C. Greubel_, Apr 25 2019

%o (Sage) ((1+x)*(1-x^3)/(1-3*x+5*x^3-3*x^4)).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 25 2019

%Y Cf. similar sequences listed in A265055.

%K nonn,easy

%O 0,2

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