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A163316 Number of reduced words of length n in Coxeter group on 5 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

%I A163316 #17 Sep 08 2022 08:45:46
%S A163316 1,5,20,80,320,1270,5040,20010,79440,315360,1251930,4969980,19730070,
%T A163316 78325380,310939920,1234384470,4900319640,19453527810,77227563240,
%U A163316 306581745960,1217083163130,4831636082580,19180864497870,76145131089180
%N A163316 Number of reduced words of length n in Coxeter group on 5 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.
%C A163316 The initial terms coincide with those of A003947, although the two sequences are eventually different.
%C A163316 Computed with MAGMA using commands similar to those used to compute A154638.
%H A163316 G. C. Greubel, <a href="/A163316/b163316.txt">Table of n, a(n) for n = 0..1000</a>
%H A163316 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3, 3, 3, 3, -6).
%F A163316 G.f.: (t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(6*t^5 - 3*t^4 - 3*t^3 - 3*t^2 - 3*t + 1).
%F A163316 a(n) = 3*a(n-1)+3*a(n-2)+3*a(n-3)+3*a(n-4)-6*a(n-5). - _Wesley Ivan Hurt_, May 10 2021
%t A163316 CoefficientList[Series[(1+x)*(1-x^5)/(1-4*x+9*x^5-6*x^6), {x, 0, 30}], x] (* or *) LinearRecurrence[{3,3,3,3,-6}, {1,5,20,80,320,1270}, 30] (* _G. C. Greubel_, Dec 18 2016 *)
%t A163316 coxG[{5, 6, -3}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, May 12 2019 *)
%o A163316 (PARI) my(x='x+O('x^30)); Vec((1+x)*(1-x^5)/(1-4*x+9*x^5-6*x^6)) \\ _G. C. Greubel_, Dec 18 2016
%o A163316 (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^5)/(1-4*x+9*x^5-6*x^6) )); // _G. C. Greubel_, May 12 2019
%o A163316 (Sage) ((1+x)*(1-x^5)/(1-4*x+9*x^5-6*x^6)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, May 12 2019
%K A163316 nonn
%O A163316 0,2
%A A163316 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009