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A163878 Number of reduced words of length n in Coxeter group on 5 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

%I A163878 #20 Sep 08 2022 08:45:47
%S A163878 1,5,20,80,320,1280,5110,20400,81450,325200,1298400,5184000,20697690,
%T A163878 82637820,329940630,1317324420,5259563280,20999387520,83842374870,
%U A163878 334749945240,1336526142210,5336228292840,21305481048360,85064487085440
%N A163878 Number of reduced words of length n in Coxeter group on 5 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.
%C A163878 The initial terms coincide with those of A003947, although the two sequences are eventually different.
%C A163878 Computed with MAGMA using commands similar to those used to compute A154638.
%H A163878 G. C. Greubel, <a href="/A163878/b163878.txt">Table of n, a(n) for n = 0..1000</a>
%H A163878 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,3,3,3,3,-6).
%F A163878 G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(6*t^6 - 3*t^5 - 3*t^4 - 3*t^3 - 3*t^2 - 3*t + 1).
%F A163878 a(n) = -6*a(n-6) + 3*Sum_{k=1..5} a(n-k). - _Wesley Ivan Hurt_, May 11 2021
%p A163878 seq(coeff(series((1+t)*(1-t^6)/(1-4*t+9*t^6-6*t^7), t, n+1), t, n), n = 0 .. 30); # _G. C. Greubel_, Aug 10 2019
%t A163878 CoefficientList[Series[(1+t)*(1-t^6)/(1-4*t+9*t^6-6*t^7), {t, 0, 30}], t] (* _G. C. Greubel_, Aug 07 2017 *)
%t A163878 coxG[{6, 6, -3}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Aug 10 2019 *)
%o A163878 (PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-4*t+9*t^6-6*t^7)) \\ _G. C. Greubel_, Aug 07 2017
%o A163878 (Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-4*t+9*t^6-6*t^7) )); // _G. C. Greubel_, Aug 10 2019
%o A163878 (Sage)
%o A163878 def A163878_list(prec):
%o A163878     P.<t> = PowerSeriesRing(ZZ, prec)
%o A163878     return P((1+t)*(1-t^6)/(1-4*t+9*t^6-6*t^7)).list()
%o A163878 A163878_list(30) # _G. C. Greubel_, Aug 10 2019
%o A163878 (GAP) a:=[5,20,80,320,1280,5110];; for n in [7..30] do a[n]:=3*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -6*a[n-6]; od; Concatenation([1], a); # _G. C. Greubel_, Aug 10 2019
%K A163878 nonn
%O A163878 0,2
%A A163878 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009