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Number of reduced words of length n in Coxeter group on 34 generators S_i with relations (S_i)^2 = (S_i S_j)^7 = I.
2

%I #10 Sep 08 2022 08:45:47

%S 1,34,1122,37026,1221858,40321314,1330603362,43909910385,

%T 1449027024192,47817891187968,1577990389060800,52073682174315648,

%U 1718431489817621568,56708238440133282816,1871371844637407092464,61755270084763733187072

%N Number of reduced words of length n in Coxeter group on 34 generators S_i with relations (S_i)^2 = (S_i S_j)^7 = I.

%C The initial terms coincide with those of A170753, 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="/A164670/b164670.txt">Table of n, a(n) for n = 0..655</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (32,32,32,32,32,32,-528).

%F G.f.: (t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(528*t^7 - 32*t^6 - 32*t^5 - 32*t^4 - 32*t^3 - 32*t^2 - 32*t + 1).

%p seq(coeff(series((1+t)*(1-t^7)/(1-33*t+560*t^7-528*t^8), t, n+1), t, n), n = 0 .. 20); # _G. C. Greubel_, Sep 15 2019

%t CoefficientList[Series[(1+t)*(1-t^7)/(1-33*t+560*t^7-528*t^8), {t, 0, 20}], t] (* _G. C. Greubel_, Sep 15 2019 *)

%t coxG[{7, 528, -32}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Sep 15 2019 *)

%o (PARI) my(t='t+O('t^20)); Vec((1+t)*(1-t^7)/(1-33*t+560*t^7-528*t^8)) \\ _G. C. Greubel_, Sep 15 2019

%o (Magma) R<t>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+t)*(1-t^7)/(1-33*t+560*t^7-528*t^8) )); // _G. C. Greubel_, Sep 15 2019

%o (Sage)

%o def A164670_list(prec):

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

%o return P((1+t)*(1-t^7)/(1-33*t+560*t^7-528*t^8)).list()

%o A164670_list(20) # _G. C. Greubel_, Sep 15 2019

%o (GAP) a:=[34, 1122, 37026, 1221858, 40321314, 1330603362, 43909910385];; for n in [8..20] do a[n]:=32*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]+a[n-6]) -528*a[n-7]; od; Concatenation([1], a); # _G. C. Greubel_, Sep 15 2019

%K nonn

%O 0,2

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