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%I #18 Sep 08 2022 08:45:47
%S 1,9,72,576,4608,36864,294912,2359260,18873792,150988068,1207886400,
%T 9662946048,77302407168,618409967616,4947205424364,39577048871472,
%U 316611634855572,2532855030486480,20262535861599360,162097851871033344
%N Number of reduced words of length n in Coxeter group on 9 generators S_i with relations (S_i)^2 = (S_i S_j)^7 = I.
%C The initial terms coincide with those of A003951, 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="/A164375/b164375.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,7,7,7,7,7,-28).
%F G.f.: (t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(28*t^7 - 7*t^6 - 7*t^5 - 7*t^4 - 7*t^3 - 7*t^2 - 7*t + 1).
%F a(n) = -28*a(n-7) + 7*Sum_{k=1..6} a(n-k). - _Wesley Ivan Hurt_, May 11 2021
%p seq(coeff(series((1+t)*(1-t^7)/(1-8*t+35*t^7-28*t^8), t, n+1), t, n), n = 0 .. 30); # _G. C. Greubel_, Aug 10 2019
%t CoefficientList[Series[(1+t)*(1-t^7)/(1-8*t+35*t^7-28*t^8), {t, 0, 30}], t] (* _G. C. Greubel_, Sep 17 2017 *)
%t coxG[{7,28,-7}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, Aug 20 2019 *)
%o (PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^7)/(1-8*t+35*t^7-28*t^8)) \\ _G. C. Greubel_, Sep 17 2017
%o (Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^7)/(1-8*t+35*t^7-28*t^8) )); // _G. C. Greubel_, Aug 10 2019
%o (Sage)
%o def A164375_list(prec):
%o P.<t> = PowerSeriesRing(ZZ, prec)
%o return P((1+t)*(1-t^7)/(1-8*t+35*t^7-28*t^8)).list()
%o A164375_list(30) # _G. C. Greubel_, Aug 10 2019
%o (GAP) a:=[9, 72, 576, 4608, 36864, 294912, 2359260];; for n in [8..30] do a[n]:=7*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]+a[n-6]) -28*a[n-7]; 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
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