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A163953 Number of reduced words of length n in Coxeter group on 9 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I. 1

%I

%S 1,9,72,576,4608,36864,294876,2358720,18867492,150921792,1207229184,

%T 9656672256,77244089580,617878417968,4942433025684,39534710232528,

%U 316239654648960,2529613056079872,20234471292326844,161856307428494112

%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)^6 = 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="/A163953/b163953.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (7,7,7,7,7,-28).

%F G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(28*t^6 - 7*t^5 - 7*t^4 - 7*t^3 - 7*t^2 - 7*t + 1).

%p seq(coeff(series((1+t)*(1-t^6)/(1-8*t+35*t^6-28*t^7), t, n+1), t, n), n = 0 .. 30); # _G. C. Greubel_, Aug 10 2019

%t CoefficientList[Series[(1+t)*(1-t^6)/(1-8*t+35*t^6-28*t^7), {t, 0, 30}], t] (* _G. C. Greubel_, Aug 13 2017 *)

%t coxG[{6, 28, -7}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Aug 10 2019 *)

%o (PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-8*t+35*t^6-28*t^7)) \\ _G. C. Greubel_, Aug 13 2017

%o (MAGMA) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-8*t+35*t^6-28*t^7) )); // _G. C. Greubel_, Aug 10 2019

%o (Sage)

%o def A163953_list(prec):

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

%o return P((1+t)*(1-t^6)/(1-8*t+35*t^6-28*t^7)).list()

%o A163953_list(30) # _G. C. Greubel_, Aug 10 2019

%o (GAP) a:=[9,72,576,4608,36864,294876];; for n in [7..30] do a[n]:=7*(a[n-1]+a[n-2]+a[n-3]+a[n-4]+a[n-5]) -28*a[n-6]; 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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Last modified October 20 21:29 EDT 2019. Contains 328273 sequences. (Running on oeis4.)