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A163924 Number of reduced words of length n in Coxeter group on 8 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

%I A163924 #18 Sep 08 2022 08:45:47
%S A163924 1,8,56,392,2744,19208,134428,940800,6584256,46080384,322496832,
%T A163924 2257016832,15795891636,110548662840,773682621768,5414672451384,
%U A163924 37894967433288,265210605012024,1856095143363468,12990012903371952
%N A163924 Number of reduced words of length n in Coxeter group on 8 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.
%C A163924 The initial terms coincide with those of A003950, although the two sequences are eventually different.
%C A163924 Computed with MAGMA using commands similar to those used to compute A154638.
%H A163924 G. C. Greubel, <a href="/A163924/b163924.txt">Table of n, a(n) for n = 0..1000</a>
%H A163924 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,6,6,6,6,-21).
%F A163924 G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(21*t^6 - 6*t^5 - 6*t^4 - 6*t^3 - 6*t^2 - 6*t + 1).
%F A163924 a(n) = -21*a(n-6) + 6*Sum_{k=1..5} a(n-k). - _Wesley Ivan Hurt_, May 11 2021
%p A163924 seq(coeff(series((1+t)*(1-t^6)/(1-7*t+27*t^6-21*t^7), t, n+1), t, n), n = 0 .. 30); # _G. C. Greubel_, Aug 10 2019
%t A163924 coxG[{6,21,-6}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, Jan 24 2016 *)
%t A163924 CoefficientList[Series[(1+t)*(1-t^6)/(1-7*t+27*t^6-21*t^7), {t,0,30}], t] (* _G. C. Greubel_, Aug 08 2017 *)
%o A163924 (PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-7*t+27*t^6-21*t^7)) \\ _G. C. Greubel_, Aug 08 2017
%o A163924 (Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-7*t+27*t^6-21*t^7) )); // _G. C. Greubel_, Aug 10 2019
%o A163924 (Sage)
%o A163924 def A163924_list(prec):
%o A163924     P.<t> = PowerSeriesRing(ZZ, prec)
%o A163924     return P((1+t)*(1-t^6)/(1-7*t+27*t^6-21*t^7)).list()
%o A163924 A163924_list(30) # _G. C. Greubel_, Aug 10 2019
%o A163924 (GAP) a:=[8,56,392,2744,19208,134428];; for n in [7..30] do a[n]:=6*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -21*a[n-6]; od; Concatenation([1], a); # _G. C. Greubel_, Aug 10 2019
%K A163924 nonn
%O A163924 0,2
%A A163924 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009