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%I #19 Sep 08 2022 08:45:47
%S 1,4,12,36,108,324,972,2916,8742,26208,78576,235584,706320,2117664,
%T 6349104,19035648,57071982,171111132,513019140,1538115228,4611520836,
%U 13826093148,41452886916,124282529820,372619336494,1117173669768
%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)^8 = 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 Vincenzo Librandi, <a href="/A164697/b164697.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,2,2,2,2,2,-3).
%F G.f.: (x^8 + 2*x^7 + 2*x^6 + 2*x^5 + 2*x^4 + 2*x^3 + 2*x^2 + 2*x + 1)/( 3*x^8 - 2*x^7 - 2*x^6 - 2*x^5 - 2*x^4 - 2*x^3 - 2*x^2 - 2*x + 1).
%F a(n) = -3*a(n-8) + 2*Sum_{k=1..7} a(n-k). - _Wesley Ivan Hurt_, May 11 2021
%p seq(coeff(series((1+t)*(1-t^8)/(1-3*t+5*t^8-3*t^9), t, n+1), t, n), n = 0..30); # _G. C. Greubel_, Sep 16 2019
%t CoefficientList[Series[(x^8 +2x^7 +2x^6 +2x^5 +2x^4 +2x^3 +2x^2 +2x +1)/( 3x^8 -2x^7 -2x^6 -2x^5 -2x^4 -2x^3 -2x^2 -2x +1), {x, 0, 40}], x ] (* _Vincenzo Librandi_, Apr 29 2014 *)
%t coxG[{8,3,-2,30}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, Jan 03 2019 *)
%o (PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^8)/(1-3*t+5*t^8-3*t^9)) \\ _G. C. Greubel_, Sep 16 2019
%o (Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^8)/(1-3*t+5*t^8-3*t^9) )); // _G. C. Greubel_, Sep 16 2019
%o (Sage)
%o def A164697_list(prec):
%o P.<t> = PowerSeriesRing(ZZ, prec)
%o return P((1+t)*(1-t^8)/(1-3*t+5*t^8-3*t^9)).list()
%o A164697_list(30) # _G. C. Greubel_, Sep 16 2019
%o (GAP) a:=[4, 12, 36, 108, 324, 972, 2916, 8742];; for n in [9..30] do a[n]:=2*Sum([1..7], j-> a[n-j]) -3*a[n-8]; od; Concatenation([1], a); # _G. C. Greubel_, Sep 16 2019
%K nonn,easy
%O 0,2
%A _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009
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