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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)^3 = I.
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%I #19 Oct 07 2024 01:32:31

%S 1,9,72,540,4032,29988,223020,1658160,12328596,91662732,681510816,

%T 5067014148,37673118252,280098623952,2082525799284,15483523651596,

%U 115119584685504,855911035979748,6363675682412076,47313758657548656,351776531372292180,2615449111101347724,19445794254904116960

%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)^3 = I.

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

%C First disagreement at index 3: a(3) = 540, A003951(3) = 576. - _Klaus Brockhaus_, Jun 15 2011

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

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,7,-28).

%F G.f.: (t^3 + 2*t^2 + 2*t + 1)/(28*t^3 - 7*t^2 - 7*t + 1)

%F a(0)=1, a(1)=9, a(2)=72, a(3)=540, a(n)=7*a(n-1)+7*a(n-2)-28*a(n-3). - _Harvey P. Dale_, Jun 15 2011

%t Join[{1},LinearRecurrence[{7,7,-28},{9,72,540},50]] (* or *) CoefficientList[ Series[(t^3+2t^2+2t+1)/(28t^3-7t^2-7t+1),{t,0,50}],t] (* _Harvey P. Dale_, Jun 15 2011 *)

%Y Cf. A003951 (G.f.: (1+x)/(1-8*x)).

%K nonn,easy

%O 0,2

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