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A162879 Number of reduced words of length n in Coxeter group on 42 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I.

%I A162879 #22 Sep 08 2022 08:45:46
%S A162879 1,42,1722,69741,2824080,114340800,4629407580,187434189600,
%T A162879 7588784431200,307252630616400,12439960566432000,503665724648352000,
%U A162879 20392280251485912000,825637071380896320000,33428168171083640640000
%N A162879 Number of reduced words of length n in Coxeter group on 42 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I.
%C A162879 The initial terms coincide with those of A170761, although the two sequences are eventually different.
%C A162879 Computed with MAGMA using commands similar to those used to compute A154638.
%H A162879 Vincenzo Librandi, <a href="/A162879/b162879.txt">Table of n, a(n) for n = 0..500</a>
%H A162879 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (40, 40, -820).
%F A162879 G.f.: (t^3 + 2*t^2 + 2*t + 1)/(820*t^3 - 40*t^2 - 40*t + 1).
%F A162879 a(n) = 40*a(n-1) + 40*a(n-2) - 820*a(n-3), n > 0. - _Muniru A Asiru_, Oct 24 2018
%F A162879 G.f.: (1+x)*(1-x^3)/(1 - 41*x + 860*x^3 - 820*x^4). - _G. C. Greubel_, Apr 27 2019
%p A162879 seq(coeff(series((x^3+2*x^2+2*x+1)/(820*x^3-40*x^2-40*x+1),x,n+1), x, n), n = 0 .. 20); # _Muniru A Asiru_, Oct 24 2018
%t A162879 CoefficientList[Series[(t^3+2*t^2+2*t+1)/(820*t^3-40*t^2-40*t+1), {t, 0, 20}], t] (* _Wesley Ivan Hurt_, Apr 12 2017 *)
%t A162879 Join[{1}, LinearRecurrence[{40, 40, -820}, {42, 1722, 69741}, 20]] (* _Vincenzo Librandi_, Apr 14 2017 *)
%t A162879 coxG[{3, 820, -40}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Apr 27 2019 *)
%o A162879 (Magma) I:=[1,42,1722,69741]; [n le 4 select I[n] else 40*Self(n-1) +40*Self(n-2)-820*Self(n-3): n in [1..20]]; // _Vincenzo Librandi_, Apr 14 2017
%o A162879 (Magma) R<t>:=PowerSeriesRing(Integers(), 20); Coefficients(R!((t^3 +2*t^2+2*t+1)/(820*t^3-40*t^2-40*t+1))); // _G. C. Greubel_, Oct 24 2018
%o A162879 (PARI) my(t='t+O('t^20)); Vec((t^3+2*t^2+2*t+1)/(820*t^3-40*t^2-40*t+1)) \\ _G. C. Greubel_, Oct 24 2018
%o A162879 (GAP) a:=[42,1722,69741];; for n in [4..20] do a[n]:=40*a[n-1]+40*a[n-2] -820*a[n-3]; od; Concatenation([1],a); # _Muniru A Asiru_, Oct 24 2018
%o A162879 (Sage) ((1+x)*(1-x^3)/(1 -41*x +860*x^3 -820*x^4)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 27 2019
%Y A162879 Cf. A154638, A170761.
%K A162879 nonn
%O A162879 0,2
%A A162879 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009