A162783 - OEIS

%I #12 Sep 08 2022 08:45:46

%S 1,14,182,2275,28392,353808,4408950,54938520,684572616,8530235532,

%T 106292493216,1324476080928,16503864518232,205649272719072,

%U 2562528512535264,31930831990629936,397879682765894784

%N Number of reduced words of length n in Coxeter group on 14 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I.

%C The initial terms coincide with those of A170733, although the two sequences are eventually different.

%C Computed with MAGMA using commands similar to those used to compute A154638.

%H G. C. Greubel, <a href="/A162783/b162783.txt">Table of n, a(n) for n = 0..900</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (12, 12, -78).

%F G.f.: (t^3 + 2*t^2 + 2*t + 1)/(78*t^3 - 12*t^2 - 12*t + 1).

%F G.f.: (1+x)*(1-x^3)/(1 - 13*x + 90*x^3 - 78*x^4). - _G. C. Greubel_, Apr 26 2019

%t CoefficientList[Series[(1+x)*(1-x^3)/(1-13*x+90*x^3-78*x^4), {x,0,20}],x] (* or *) coxG[{3, 78, -12}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Apr 26 2019 *)

%o (PARI) my(x='x+O('x^20)); Vec((1+x)*(1-x^3)/(1-13*x+90*x^3-78*x^4)) \\ _G. C. Greubel_, Apr 26 2019

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^3)/(1-13*x+90*x^3-78*x^4) )); // _G. C. Greubel_, Apr 26 2019

%o (Sage) ((1+x)*(1-x^3)/(1-13*x+90*x^3-78*x^4)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 26 2019

%o (GAP) a:=[14, 182, 2275];; for n in [4..20] do a[n]:=12*a[n-1]+12*a[n-2] - 78*a[n-3]; od; Concatenation([1], a); # _G. C. Greubel_, Apr 26 2019

%K nonn

%O 0,2

%A _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009