%I #16 Sep 08 2022 08:45:48
%S 1,4,12,36,108,324,972,2916,8748,26244,78726,236160,708432,2125152,
%T 6375024,19123776,57367440,172090656,516236976,1548605952,4645502958,
%U 13935564252,41803859076,125403076764,376183730628,1128474698076
%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)^10 = 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 G. C. Greubel, <a href="/A165756/b165756.txt">Table of n, a(n) for n = 0..500</a>
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,2,2,2,2,2,2,2,-3).
%F G.f.: (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^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).
%p seq(coeff(series((1+t)*(1-t^10)/(1-3*t+5*t^10-3*t^11), t, n+1), t, n), n = 0 .. 30); # _G. C. Greubel_, Sep 16 2019
%t CoefficientList[Series[(1+t)*(1-t^10)/(1-3*t+5*t^10-3*t^11), {t, 0, 30}], t] (* _G. C. Greubel_, Apr 07 2016 *)
%t coxG[{10, 3, -2}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Sep 16 2019 *)
%o (PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^10)/(1-3*t+5*t^10-3*t^11)) \\ _G. C. Greubel_, Sep 16 2019
%o (Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^10)/(1-3*t+5*t^10-3*t^11) )); // _G. C. Greubel_, Sep 16 2019
%o (Sage)
%o def A165756_list(prec):
%o P.<t> = PowerSeriesRing(ZZ, prec)
%o return P((1+t)*(1-t^10)/(1-3*t+5*t^10-3*t^11)).list()
%o A165756_list(30) # _G. C. Greubel_, Sep 16 2019
%o (GAP) a:=[4, 12, 36, 108, 324, 972, 2916, 8748, 26244, 78726];; for n in [11..30] do a[n]:=2*Sum([1..9], j-> a[n-j]) -3*a[n-10]; od; Concatenation([1], a); # _G. C. Greubel_, Sep 16 2019
%K nonn
%O 0,2
%A _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009
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