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%I #20 Sep 08 2022 08:45:46
%S 1,38,1406,51319,1872792,68331600,2493179658,90967125816,
%T 3319062151464,121100596329852,4418523599533920,161215975658220768,
%U 5882188976123487336,214619841546851901024,7830703259038738949472
%N Number of reduced words of length n in Coxeter group on 38 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I.
%C The initial terms coincide with those of A170757, 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="/A162858/b162858.txt">Table of n, a(n) for n = 0..638</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (36, 36, -666).
%F G.f.: (t^3 + 2*t^2 + 2*t + 1)/(666*t^3 - 36*t^2 - 36*t + 1).
%F a(n) = 36*a(n-1) + 36*a(n-2) - 666*a(n-3), n > 0. - _Muniru A Asiru_, Oct 25 2018
%F G.f.: (1+x)*(1-x^3)/(1 - 37*x + 702*x^3 - 666*x^4). - _G. C. Greubel_, Apr 27 2019
%p seq(coeff(series((x^3+2*x^2+2*x+1)/(666*x^3-36*x^2-36*x+1),x,n+1), x, n), n = 0 .. 20); # _Muniru A Asiru_, Oct 25 2018
%t CoefficientList[Series[(t^3+2*t^2+2*t+1)/(666*t^3-36*t^2-36*t+1), {t, 0, 20}], t] (* _G. C. Greubel_, Oct 24 2018 *)
%t coxG[{3, 666, -36}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Apr 27 2019 *)
%o (PARI) my(t='t+O('t^20)); Vec((t^3+2*t^2+2*t+1)/(666*t^3-36*t^2-36*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)/(666*t^3-36*t^2-36*t+1))); // _G. C. Greubel_, Oct 24 2018
%o (GAP) a:=[38,1406,51319];; for n in [4..20] do a[n]:=36*a[n-1]+36*a[n-2]-666*a[n-3]; od; Concatenation([1],a); # _Muniru A Asiru_, Oct 25 2018
%o (Sage) ((1+x)*(1-x^3)/(1 -37*x +702*x^3 -666*x^4)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 26 2019
%K nonn
%O 0,2
%A _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009
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