|
%I #17 Mar 24 2021 08:05:34
%S 1,8,56,392,2744,19208,134456,941192,6588344,46118408,322828856,
%T 2259801992,15818613944,110730297608,775112083256,5425784582792,
%U 37980492079544,265863444556780,1861044111897264,13027308783279504
%N Number of reduced words of length n in Coxeter group on 8 generators S_i with relations (S_i)^2 = (S_i S_j)^17 = I.
%C The initial terms coincide with those of A003950, although the two sequences are eventually different.
%C First disagreement at index 17: a(17) = 265863444556780, A003950(17) = 265863444556808. - _Klaus Brockhaus_, Mar 30 2011
%C Computed with MAGMA using commands similar to those used to compute A154638.
%H G. C. Greubel, <a href="/A168685/b168685.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 (6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,-21).
%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)/ (21*t^17 - 6*t^16 - 6*t^15 - 6*t^14 - 6*t^13 - 6*t^12 - 6*t^11 - 6*t^10 - 6*t^9 - 6*t^8 - 6*t^7 - 6*t^6 - 6*t^5 - 6*t^4 - 6*t^3 - 6*t^2 - 6*t + 1).
%F G.f.: (1+t)*(1-t^17)/(1 - 7*t + 27*t^17 - 21*t^18). - _G. C. Greubel_, Mar 24 2021
%t CoefficientList[Series[(1+t)*(1-t^17)/(1 -7*t +27*t^17 -21*t^18), {t, 0, 40}], t] (* _G. C. Greubel_, Aug 03 2016; Mar 24 2021 *)
%t coxG[{17, 21, -6, 40}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Mar 24 2021 *)
%o (Magma)
%o R<t>:=PowerSeriesRing(Integers(), 40);
%o Coefficients(R!( (1+t)*(1-t^17)/(1 -7*t +27*t^17 -21*t^18) )); // _G. C. Greubel_, Mar 24 2021
%o (Sage)
%o def A168685_list(prec):
%o P.<t> = PowerSeriesRing(ZZ, prec)
%o return P( (1+t)*(1-t^17)/(1 -7*t +27*t^17 -21*t^18) ).list()
%o A168685_list(40) # _G. C. Greubel_, Mar 24 2021
%Y Cf. A003950 (g.f.: (1+x)/(1-7*x)).
%K nonn
%O 0,2
%A _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009
| 