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%I #19 Sep 08 2022 08:45:48
%S 1,4,12,36,108,324,972,2916,8748,26244,78732,236196,708582,2125728,
%T 6377136,19131264,57393360,172178784,516532464,1549585728,4648722192,
%U 13946061600,41837869872,125512664832,376535160174,1129596977628
%N Number of reduced words of length n in Coxeter group on 4 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.
%C The initial terms coincide with those of A003946, although the two sequences are eventually different.
%C Computed with MAGMA using commands similar to those used to compute A154638.
%H Vincenzo Librandi, <a href="/A166468/b166468.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, -3).
%F G.f.: (t^12 + 2*t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(3*t^12 - 2*t^11 - 2*t^10 - 2*t^9 - 2*t^8 - 2*t^7 - 2*t^6 - 2*t^5 - 2*t^4 - 2*t^3 - 2*t^2 - 2*t + 1).
%F G.f.: (1+x)*(1-x^12)/(1 -3*x +5*x^12 -3*x^13). - _G. C. Greubel_, Apr 26 2019
%F a(n) = -3*a(n-12) + 2*Sum_{k=1..11} a(n-k). - _Wesley Ivan Hurt_, May 06 2021
%t CoefficientList[Series[(1+x)*(1-x^12)/(1 -3*x +5*x^12 -3*x^13), {x, 0, 30}], x ] (* _Vincenzo Librandi_, Apr 29 2014 *)(* modified by _G. C. Greubel_, Apr 26 2019 *)
%t coxG[{12,3,-2,30}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, May 09 2018 *)
%o (PARI) my(x='x+O('x^30)); Vec((1+x)*(1-x^12)/(1-3*x+5*x^12-3*x^13)) \\ _G. C. Greubel_, Apr 26 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^12)/(1-3*x+5*x^12-3*x^13) )); // _G. C. Greubel_, Apr 26 2019
%o (Sage) ((1+x)*(1-x^12)/(1-3*x+5*x^12-3*x^13)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 26 2019
%K nonn,easy
%O 0,2
%A _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009
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