A162851 - OEIS

%I #18 Sep 08 2022 08:45:46

%S 1,37,1332,47286,1678320,59557050,2113447770,74997827100,

%T 2661373678950,94441530616650,3351353019273000,118926143828399250,

%U 4220214225380039250,149758560520153357500,5314333645481777358750,188584492248078150341250

%N Number of reduced words of length n in Coxeter group on 37 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I.

%C The initial terms coincide with those of A170756, 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="/A162851/b162851.txt">Table of n, a(n) for n = 0..643</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (35, 35, -630).

%F G.f.: (t^3 + 2*t^2 + 2*t + 1)/(630*t^3 - 35*t^2 - 35*t + 1).

%F G.f.: (1+x)*(1-x^3)/(1 - 36*x + 665*x^3 - 630*x^4). - _G. C. Greubel_, Apr 26 2019

%F a(n) = 35*a(n-1)+35*a(n-2)-630*a(n-3). - _Wesley Ivan Hurt_, May 05 2021

%t CoefficientList[Series[(t^3+2*t^2+2*t+1)/(630*t^3-35*t^2-35*t+1), {t, 0, 20}], t] (* or *) LinearRecurrence[{35, 35, -630}, {1, 37, 1332}, 20] (* _G. C. Greubel_, Oct 24 2018 *)

%t coxG[{3, 630, -35}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Apr 26 2019 *)

%o (PARI) my(t='t+O('t^20)); Vec((t^3+2*t^2+2*t+1)/(630*t^3-35*t^2-35*t+1)) \\ _G. C. Greubel_, Oct 24 2018

%o (Magma) R<t>:=PowerSeriesRing(Integers(), 20); Coefficients(R!((t^3 +2*t^2+2*t+1)/(630*t^3-35*t^2-35*t+1))); // _G. C. Greubel_, Oct 24 2018

%o (Sage) ((1+x)*(1-x^3)/(1-36*x+665*x^3-630*x^4)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 26 2019

%o (GAP) a:=[37, 1332, 47286];; for n in [4..20] do a[n]:=35*a[n-1]+ 35*a[n-2]-630*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