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A163314 Number of reduced words of length n in Coxeter group on 3 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

%I A163314 #20 Sep 08 2022 08:45:46
%S A163314 1,3,6,12,24,45,84,159,300,564,1062,2001,3768,7095,13362,25164,47388,
%T A163314 89241,168060,316491,596016,1122420,2113746,3980613,7496304,14117067,
%U A163314 26585310,50065548,94283616,177555237,334372644,629691735,1185837684
%N A163314 Number of reduced words of length n in Coxeter group on 3 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.
%C A163314 The initial terms coincide with those of A003945, although the two sequences are eventually different.
%C A163314 Computed with MAGMA using commands similar to those used to compute A154638.
%H A163314 G. C. Greubel, <a href="/A163314/b163314.txt">Table of n, a(n) for n = 0..1000</a>
%H A163314 M. O'Keeffe, <a href="https://doi.org/10.1524/zkri.1998.213.3.135">Coordination sequences for hyperbolic tilings</a>, Zeitschrift für Kristallographie, 213 (1998), 135-140 (see next-to-last table, row {10, 3}).
%H A163314 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2, -1, 2, -1).
%F A163314 G.f.: (t^4 + t^3 + t^2 + t + 1)/(t^4 - 2*t^3 + t^2 - 2*t + 1).
%F A163314 a(n) = 2*a(n-1)-a(n-2)+2*a(n-3)-a(n-4). - _Wesley Ivan Hurt_, May 10 2021
%t A163314 CoefficientList[Series[(t^4+t^3+t^2+t+1)/(t^4-2*t^3+t^2-2*t+1), {t,0,40} ], t] (* or *) LinearRecurrence[{2,-1,2,-1}, {1,3,6,12,24}, 40] (* _G. C. Greubel_, Dec 18 2016 *)
%o A163314 (PARI) my(t='t+O('t^40)); Vec((t^4+t^3+t^2+t+1)/(t^4-2*t^3+t^2-2*t+1)) \\ _G. C. Greubel_, Dec 18 2016
%o A163314 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (x^4+x^3 +x^2+x+1)/(x^4-2*x^3+x^2-2*x+1) )); // _G. C. Greubel_, May 12 2019
%o A163314 (Sage) ((x^4+x^3 +x^2+x+1)/(x^4-2*x^3+x^2-2*x+1)).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, May 12 2019
%o A163314 (GAP) a:=[3,6,12,24];; for n in [5..40] do a[n]:=2*a[n-1]-a[n-2]+ 2*a[n-3]-a[n-4]; od; Concatenation([1], a); # _G. C. Greubel_, May 12 2019
%K A163314 nonn
%O A163314 0,2
%A A163314 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009