A168683 - OEIS

%I #21 Feb 22 2021 09:19:33

%S 1,6,30,150,750,3750,18750,93750,468750,2343750,11718750,58593750,

%T 292968750,1464843750,7324218750,36621093750,183105468750,

%U 915527343735,4577636718600,22888183592640,114440917961400,572204589798000

%N Number of reduced words of length n in Coxeter group on 6 generators S_i with relations (S_i)^2 = (S_i S_j)^17 = I.

%C The initial terms coincide with those of A003948, although the two sequences are eventually different.

%C First disagreement at index 17: a(17) = 915527343735, A003948(17) = 915527343750. - _Klaus Brockhaus_, Mar 30 2011

%C Computed with MAGMA using commands similar to those used to compute A154638.

%H G. C. Greubel, <a href="/A168683/b168683.txt">Table of n, a(n) for n = 0..500</a>

%H <a href="/index/Rec#order_17">Index entries for linear recurrences with constant coefficients</a>, signature (4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,-10).

%F G.f.: (t^17 + 2*t^16 + 2*t^15 + 2*t^14 + 2*t^13 + 2*t^12 + 2*t^11 + 2*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) / (10*t^17 - 4*t^16 - 4*t^15 - 4*t^14 - 4*t^13 - 4*t^12 - 4*t^11 - 4*t^10 - 4*t^9 - 4*t^8 - 4*t^7 - 4*t^6 - 4*t^5 - 4*t^4 - 4*t^3 - 4*t^2 - 4*t + 1).

%F G.f.: (1+t)*(1-t^17)/(1 -5*t +14*t^17 -10*t^18). - _G. C. Greubel_, Feb 22 2021

%t CoefficientList[Series[(1+t)*(1-t^17)/(1 -5*t +14*t^17 -10*t^18), {t, 0, 40}], t] (* _G. C. Greubel_, Aug 03 2016, Feb 22 2021 *)

%t coxG[{17,10,-4,30}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, Jun 09 2017 *)

%o (Magma)

%o R<t>:=PowerSeriesRing(Integers(), 40);

%o Coefficients(R!( (1+t)*(1-t^17)/(1 -5*t +14*t^17 -10*t^18) )); // _G. C. Greubel_, Feb 22 2021

%o (Sage)

%o def A168683_list(prec):

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

%o     return P( (1+t)*(1-t^17)/(1 -5*t +14*t^17 -10*t^18) ).list()

%o A168683_list(40) # _G. C. Greubel_, Feb 22 2021

%Y Cf. A003948 (G.f.: (1+x)/(1-5*x)).

%K nonn,easy

%O 0,2

%A _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009