|
%I #17 Sep 08 2022 08:45:47
%S 1,11,110,1100,11000,110000,1099945,10998900,109983555,1099781100,
%T 10997266500,109967220000,1099617752970,10995633086625,
%U 109950886704780,1099454428128375,10993999919042250,109934555837535000
%N Number of reduced words of length n in Coxeter group on 11 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.
%C The initial terms coincide with those of A003953, 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="/A163955/b163955.txt">Table of n, a(n) for n = 0..995</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (9,9,9,9,9,-45).
%F G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(45*t^6 - 9*t^5 - 9*t^4 - 9*t^3 - 9*t^2 - 9*t + 1).
%F a(n) = -45*a(n-6) + 9*Sum_{k=1..5} a(n-k). - _Wesley Ivan Hurt_, May 11 2021
%p seq(coeff(series((1+t)*(1-t^6)/(1-10*t+54*t^6-45*t^7), t, n+1), t, n), n = 0 .. 30); # _G. C. Greubel_, Aug 10 2019
%t CoefficientList[Series[(1+t)*(1-t^6)/(1-10*t+54*t^6-45*t^7), {t,0,30}], t] (* _G. C. Greubel_, Aug 13 2017 *)
%t coxG[{6, 45, -9}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Aug 10 2019 *)
%o (PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-10*t+54*t^6-45*t^7)) \\ _G. C. Greubel_, Aug 13 2017
%o (Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-10*t+54*t^6-45*t^7) )); // _G. C. Greubel_, Aug 10 2019
%o (Sage)
%o def A163955_list(prec):
%o P.<t> = PowerSeriesRing(ZZ, prec)
%o return P((1+t)*(1-t^6)/(1-10*t+54*t^6-45*t^7)).list()
%o A163955_list(30) # _G. C. Greubel_, Aug 10 2019
%o (GAP) a:=[11,110,1100,11000,110000,1099945];; for n in [7..30] do a[n]:=9*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -45*a[n-6]; od; Concatenation([1], a); # _G. C. Greubel_, Aug 10 2019
%K nonn
%O 0,2
%A _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009
| 