A163954 - OEIS

%I #15 Sep 08 2022 08:45:47

%S 1,10,90,810,7290,65610,590445,5313600,47818800,430336800,3872739600,

%T 34852032000,313644670380,2822589491040,25401392681760,

%U 228595320793440,2057202978723360,18513432737727840,166608348947205840

%N Number of reduced words of length n in Coxeter group on 10 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

%C The initial terms coincide with those of A003952, 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="/A163954/b163954.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 (8,8,8,8,8,-36).

%F G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(36*t^6 - 8*t^5 - 8*t^4 - 8*t^3 - 8*t^2 - 8*t + 1).

%F a(n) = -36*a(n-6) + 8*Sum_{k=1..5} a(n-k). - _Wesley Ivan Hurt_, May 11 2021

%p seq(coeff(series((1+t)*(1-t^6)/(1-9*t+44*t^6-36*t^7), t, n+1), t, n), n = 0 .. 30); # _G. C. Greubel_, Aug 10 2019

%t CoefficientList[Series[(1+t)*(1-t^6)/(1-9*t+44*t^6-36*t^7), {t, 0, 30}], t] (* _G. C. Greubel_, Aug 13 2017 *)

%t coxG[{6, 36, -8}] (* 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)/(1-9*t+44*t^6-36*t^7)) \\ _G. C. Greubel_, Aug 13 2017

%o (Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-9*t+44*t^6-36*t^7) )); // _G. C. Greubel_, Aug 10 2019

%o (Sage)

%o def A163954_list(prec):

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

%o     return P((1+t)*(1-t^6)/(1-9*t+44*t^6-36*t^7)).list()

%o A163954_list(30) # _G. C. Greubel_, Aug 10 2019

%o (GAP) a:=[10, 90, 810, 7290, 65610, 590445];; for n in [7..30] do a[n]:=8*(a[n-1]+a[n-2]+a[n-3]+a[n-4]+a[n-5]) -36*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