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A163218 Number of reduced words of length n in Coxeter group on 35 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.

%I A163218 #14 Sep 08 2022 08:45:46
%S A163218 1,35,1190,40460,1375045,46731300,1588176975,53974651500,
%T A163218 1834344072330,62340711467265,2118667029023160,72003509011079415,
%U A163218 2447059985777227590,83164038200838759780,2826353783752411211145,96054447135432681999180
%N A163218 Number of reduced words of length n in Coxeter group on 35 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.
%C A163218 The initial terms coincide with those of A170754, although the two sequences are eventually different.
%C A163218 Computed with MAGMA using commands similar to those used to compute A154638.
%H A163218 G. C. Greubel, <a href="/A163218/b163218.txt">Table of n, a(n) for n = 0..650</a>
%H A163218 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (33, 33, 33, -561).
%F A163218 G.f.: (t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(561*t^4 - 33*t^3 - 33*t^2 - 33*t + 1).
%F A163218 a(n) = -561*a(n-4) + 33*Sum_{k=1..3} a(n-k). - _Wesley Ivan Hurt_, May 05 2021
%t A163218 CoefficientList[Series[(t^4+2*t^3+2*t^2+2*t+1)/(561*t^4-33*t^3-33*t^2 - 33*t+1), {t,0,20}], t] (* or *) LinearRecurrence[{33, 33, 33, -561}, {1, 35, 1190, 40460}, 20] (* _G. C. Greubel_, Dec 11 2016 *)
%t A163218 coxG[{4, 561, -33}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Apr 30 2019 *)
%o A163218 (PARI) my(t='t+O('t^20)); Vec((t^4+2*t^3+2*t^2+2*t+1)/(561*t^4-33*t^3 - 33*t^2-33*t+1)) \\ _G. C. Greubel_, Dec 11 2016
%o A163218 (Magma) R<x>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^4)/(1-34*x+594*x^4-x^561*x^5) )); // _G. C. Greubel_, Apr 30 2019
%o A163218 (Sage) ((1+x)*(1-x^4)/(1-34*x+594*x^4-561*x^5)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 30 2019
%K A163218 nonn
%O A163218 0,2
%A A163218 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009