A163878 - OEIS

%I #20 Sep 08 2022 08:45:47

%S 1,5,20,80,320,1280,5110,20400,81450,325200,1298400,5184000,20697690,

%T 82637820,329940630,1317324420,5259563280,20999387520,83842374870,

%U 334749945240,1336526142210,5336228292840,21305481048360,85064487085440

%N Number of reduced words of length n in Coxeter group on 5 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

%C The initial terms coincide with those of A003947, 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="/A163878/b163878.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,3,3,3,3,-6).

%F G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(6*t^6 - 3*t^5 - 3*t^4 - 3*t^3 - 3*t^2 - 3*t + 1).

%F a(n) = -6*a(n-6) + 3*Sum_{k=1..5} a(n-k). - _Wesley Ivan Hurt_, May 11 2021

%p seq(coeff(series((1+t)*(1-t^6)/(1-4*t+9*t^6-6*t^7), t, n+1), t, n), n = 0 .. 30); # _G. C. Greubel_, Aug 10 2019

%t CoefficientList[Series[(1+t)*(1-t^6)/(1-4*t+9*t^6-6*t^7), {t, 0, 30}], t] (* _G. C. Greubel_, Aug 07 2017 *)

%t coxG[{6, 6, -3}] (* 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)/(1-4*t+9*t^6-6*t^7)) \\ _G. C. Greubel_, Aug 07 2017

%o (Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-4*t+9*t^6-6*t^7) )); // _G. C. Greubel_, Aug 10 2019

%o (Sage)

%o def A163878_list(prec):

%o     P.<t> = PowerSeriesRing(ZZ, prec)

%o     return P((1+t)*(1-t^6)/(1-4*t+9*t^6-6*t^7)).list()

%o A163878_list(30) # _G. C. Greubel_, Aug 10 2019

%o (GAP) a:=[5,20,80,320,1280,5110];; for n in [7..30] do a[n]:=3*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -6*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